## This is PART 3: Centers X(3001) - X(5000)

 PART 1: Introduction and Centers X(1) - X(1000) PART 2: Centers X(1001) - X(3000) PART 3: Centers X(3001) - X(5000) PART 4: Centers X(5001) - X(7000) PART 5: Centers X(7001) - X(10000) PART 6: Centers X(10001) - X(12000) PART 7: Centers X(12001) - X(14000) PART 8: Centers X(14001) - X(16000) PART 9: Centers X(16001) - X(18000) PART 10: Centers X(18001) - X(20000) PART 11: Centers X(20001) - X(22000) PART 12: Centers X(22001) - X(24000) PART 13: Centers X(24001) - X(26000) PART 14: Centers X(26001) - X(28000) PART 15: Centers X(28001) - X(30000) PART 16: Centers X(30001) - X(32000) PART 17: Centers X(32001) - X(34000) PART 18: Centers X(34001) - X(36000) PART 19: Centers X(36001) - X(38000) PART 20: Centers X(38001) - X(40000)

### X(3001) = (BROCARD AXIS)∩(DE LONGCHAMPS LINE)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b6 + c6 - a2b4 - a2c4)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3001) lies on these lines: 3,6   325,523   1634,2393   1990,2967

X(3001) = isotomic conjugate of X(2367)

### X(3002) = (BROCARD AXIS)∩(GERGONNE LINE)

Trilinears        f(A,B,C)) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin(A - B) cos2(B/2) + sin(A - C) cos2(C/2)
Barycentrics   (sin A)f(A,B,C)) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3002) lies on these lines: 3,6   36,1951   241,514   978,1047   1758,1945

X(3002) = crosssum of X(6) and X(851)

### X(3003) = (BROCARD AXIS)∩(ORTHIC AXIS)

Trilinears    a[(a2 - b2)sin 2B + (a2 - c2)sin 2C] : :
Trilinears    sin A - sin 2A cos(B - C) : :      (Joe Goggins, 11/26/08)

Trilinears    (sin A)(1 + cos 2B + cos 2C) : :

X(3003) lies on these lines: 3,6   53,235   157,1974   230,231   237,2393   248,1177   393,847   419,1632   607,2178   608,2164   1100,2649   1942,1987   2174,2197

X(3003) = isogonal conjugate of X(2986)
X(3003) = complement of X(3260)
X(3003) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,113), (186,1495), (476,512), (687,924), (1299,25), (2407,526)
X(3003) = crosspoint of X(i) and X(j) for these (i,j): (2,74), (6,1989), (249,1304)
X(3003) = crosssum of X(i) and X(j) for these (i,j): (2,323), (6,30)
X(3003) = perspector of circumconic centered at X(113)
X(3003) = center of circumconic that is locus of trilinear poles of lines passing through X(113)
X(3003) = Brocard axis intercept of line through X(371)-Ceva conjugate of X(372) and X(372)-Ceva conjugate of X(371)
X(3003) = center of bicevian conic of X(15) and X(16)
X(3003) = crossdifference of every pair of points on line X(3)X(523)

### X(3004) = (DE LONGCHAMPS LINE)∩(GERGONNE LINE)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b - c)(b2 + c2 + ab + ac)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3004) lies on these lines: 241,514   325,523   522,2526   661,918

X(3004) = isotomic conjugate of X(8707)
X(3004) = X(i)-Ceva conjugate of X(j) for these (i,j): (274,1086), (1014,1565), (1441,1111)
X(3004) = crosssum of X(6) and X(2483)

### X(3005) = (DE LONGCHAMPS LINE)∩(LEMOINE AXIS)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b4 - c4)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let L denote the line at infinity. Then (Lemoine axis) = (isogonal conjugate of isotomic conjugate of L) and (de Longchamps line) = (isotomic conjugate of isogonal conjugate of L), and X(3005) = (Lemoine axis)∩(de Longchamps line). (Randy Hutson, February 10, 2016)

X(3005) lies on the bicevian conic of X(2) and X(512) and these lines: 2,881   187,237   325,523   661,756   689,783   826,2474   878,2623

X(3005) = reflection of X(i) in X(j) for these (i,j): (669,647), (2528,2525)
X(3005) = isogonal conjugate of X(4577)
X(3005) = isotomic conjugate of X(689)
X(3005) = X(i)-Ceva conjugate of X(j) for these (i,j): (512,688), (523,826), (827,6), (1634,39)
X(3005) = X(2531)-cross conjugate of X(688)
X(3005) = crosspoint of X(i) and X(j) for these (i,j): (6,827), (39,1634), (512,523)
X(3005) = crosssum of X(i) and X(j) for these (i,j): (2,826), (99,110), (512,1194)
X(3005) = complementary conjugate of X(7668)
X(3005) = crossdifference of every pair of points on line X(2)X(32)
X(3005) = perspector of circumconic centered at X(3124)
X(3005) = center of circumconic that is locus of trilinear poles of lines passing through X(3124)
X(3005) = X(2)-Ceva conjugate of X(3124)
X(3005) = radical center of {circumcircle, Brocard circle, symmedial circle}

### X(3006) = (DE LONGCHAMPS LINE)∩(NAGEL LINE)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b3 + c3 - ab2 - ac2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3006) lies on these lines: 1,2   38,2887   321,2886   325,523   752,896

X(3006) = isotomic conjugate of X(675)
X(3006) = anticomplement of X(3011)

### X(3007) = (DE LONGCHAMPS LINE)∩(SODDY LINE)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b3(sec2(A/2) - sec2(B/2)) + c3(sec2(A/2) - sec2(C/2))
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3007) lies on these lines: 1,7   5,1441   325,523   953,1305

X(3007) = isotomic conjugate of X(2370)
X(3007) = crosspoint of X(264) and X(903)
X(3007) = crosssum of X(184) and X(902)
X(3007) = anticomplement of X(8756)

### X(3008) = (GERGONNE LINE)∩(NAGEL LINE)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[(b - c)2 + 2a2 - ab - ac]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3008) lies on these lines: 1,2   44,527   57,169   101,1429   190,1266   218,226   238,516   241,514   379,1724   443,1453   536,2325   1445,1723

X(3008) = midpoint of X(i) and X(j) for these (i,j): (44,1086), (190,1266), (238,1738)
X(3008) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,3021), (666,514)
X(3008) = cevapoint of X(1279) and X(2348)
X(3008) = X(3021)-cross conugate of X(7)
X(3008) = crosssum of X(6) and X(672)
X(3008) = complement of X(3912)
X(3008) = inverse-in-Steiner-circumellipse of X(145)
X(3008) = inverse-in-Steiner-inellipse of X(1)

### X(3009) = (LEMOINE AXIS)∩(NAGEL LINE)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[a(b2 + c2) - bc(b + c)]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3009) lies on these lines: 1,2   31,172   37,1964   101,2210   187,237   192,1740   213,2308   238,660   292,672   536,2234   694,2054   741,1931   872,1100   1914,2109   2111,2116   2113,2114

X(3009) = isogonal conjugate of X(3226)
X(3009) = X(i)-Ceva conjugate of X(j) for these (i,j): (238,672), (660,649), (727,6), (1911,42)
X(3009) = crosspoint of X(i) and X(j) for these (i,j): (1,292), (6,727), (1463,1575)
X(3009) = crosssum of X(i) and X(j) for these (i,j): (1,239), (2,726)
X(3009) = crossdifference of every pair of points on line X(2)X(649)

### X(3010) = (LEMOINE AXIS)∩(SODDY LINE)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ab[sec2(A/2) - sec2(C/2)] + ac[sec2(A/2) - sec2(B/2)]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3010) lies on these lines: 1,7   31,1951   187,237

### X(3011) = (NAGEL LINE)∩(ORTHIC AXIS)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[2a3 + (b + c)(b - c + a)(b - c - a)]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3011) lies on these lines: 1,2   3,1072   11,1279   12,1104   25,225   31,226   100,1738   105,2006   111,2690   142,750   230,231   238,908   459,1068   851,2223   1447,1758   1612,1838   1836,3049

X(3011) = complement of X(3006)
X(3011) = crosspoint of X(2) and X(675)
X(3011) = crosssum of X(6) and X(674)
X(3011) = PU(4)-harmonic conjugate of X(7649)

### X(3012) = (ORTHIC AXIS)∩(SODDY LINE)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = [sec2(A/2) - sec2(B/2)] cos B + [sec2(A/2) - sec2(C/2)] cos C
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3012) lies on these lines: 1,7   230,231

### X(3013) = (ANTIORTHIC AXIS)∩(FERMAT AXIS)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin(A - B)[1 + 2 cos(2C)] + sin(A - C)[1 + 2 cos(2B)]
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3013) lies on these lines: 6,13   44,513

X(3013) = X(477)-Ceva conjugate of X(55)
X(3013) = crosssum of X(1) and X(3013)

### X(3014) = (DE LONGCHAMPS LINE)∩(FERMAT AXIS)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin3B sin(A - B)[1 + 2 cos(2C)] + sin3C sin(A - C)[1 + 2 cos(2B)]
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3014) lies on these lines: 6,13   325,523   868,2854

### X(3015) = (GERGONNE LINE)∩(FERMAT AXIS)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (1 + cos B) sin(A - B)[1 + 2 cos(2C)] + (1 + cos C) sin(A - C)[1 + 2 cos(2B)]
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3015) lies on these lines: 6,13   241,514

### X(3016) = (LEMOINE AXIS)∩(FERMAT AXIS)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = csc B sin(A - B)[1 + 2 cos(2C)] + csc C sin(A - C)[1 + 2 cos(2B)]
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3016) lies on these lines: 6,13   187,237   316,323   543,2421   1843,2971   2088,2493

X(3016) = reflection of X(2088) in X(2493)
X(3016) = crossdifference of every pair of points on line X(2)X(526)

### X(3017) = (NAGEL LINE)∩(FERMAT AXIS)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where
f(A,B,C) = (csc A - csc C) sin(A - B)[1 + 2 cos(2C)] - (csc A - csc B) sin(A - C)[1 + 2 cos(2B)]

X(3017) lies on these lines: 1,2   6,13   12,1126   30,58

X(3017) = X(58)-of-orthocentroidal triangle

### X(3018) = (ORTHIC AXIS)∩(FERMAT AXIS)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos B sin(A - B)[1 + 2 cos(2C)] + cos C sin(A - C)[1 + 2 cos(2B)]
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3018) lies on these lines: 6,13   111,1302   112,393   230,231   1249,2165   1609,2079

X(3018) = crosspoint of X(2) and X(477)

### X(3019) = (SODDY LINE)∩(FERMAT AXIS)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = sin(A - C)[1 + 2 cos(2B)][sec2(A/2) -sec2(B/2)] + sin(A - B)[1 + 2 cos(2C)][sec2(A/2) -sec2(C/2)]
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3019) lies on these lines: 1,7   6,13

### X(3020) = 4th STEVANOVIC POINT

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)[(b - c)(b2 + c2 - bc)]2

Centers X(3020)-X(3026) are points on the incircle, as noted by Milorad Stevanovic (Hyacinthos, Dec. 7-8, 2004)

X(3020) lies on the incircle and this line: 1086,3023

### X(3021) = 5th STEVANOVIC POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c - a)[2a2 - a(b + c) + (b - c)2]2
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3021) lies on the incircle and these lines: 1,1358   11,55   56,1292   354,1357   1317,2826   1361,2814   1362,2820   1364,2835   1682,3034   2775,3028   2788,3027   2795,3023   2809,3022   2836,3024

X(3021) = reflection of X(i) in X(j) for these (i,j): (8,3039), (1358,1)
X(3021) = X(7)-Ceva conjugate of X(3008)
X(3021) = crosspoint of X(7) and X(3008)
X(3021) = X(1297)-of-intouch-triangle
X(3021) = X(105)-of-Mandart-incircle-triangle
X(3021) = homothetic center of intangents triangle and reflection of extangents triangle in X(105)

### X(3022) = 6th STEVANOVIC POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b - c)2(b + c - a)3
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3022) lies on the incircle and these lines: 1,1362   11,116   12,118   33,181   56,103   65,1360   101,2291   150,497   152,388   926,2170   928,1364   950,2784   1282,1697   1317,2801   1359,2823   1361,2099   1397,2192   1682,3033   2772,3028   2774,3024   2786,3023   2809,3021

X(3022) = reflection of X(i) in X(j) for these (i,j): (8,3041), (1362,1)
X(3022) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,650), (55,657)
X(3022) = crosspoint of X(i) and X(j) for these (i,j): (7,650), (55,657), (607,663)
X(3022) = crosssum of X(i) and X(j) for these (i,j): (7,658), (55,651), (100,144), (279,934), (348,664)
X(3022) = X(99)-of-intouch-triangle
X(3022) = X(101)-of-Mandart-incircle-triangle
X(3022) = trilinear product of vertices of intangents triangle
X(3022) = trilinear product of vertices of Mandart-incircle triangle
X(3022) = homothetic center of intangents triangle and reflection of extangents triangle in X(101)
X(3022) = intersection, other than vertices of intouch triangle, of incircle and Privalov conic

### X(3023) = 7th STEVANOVIC POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c - a)(b - c)2(a2 + bc)2
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3023) lies on the incircle and these lines: 1,2782   11,115   12,114   56,98   65,1355   147,388   148,497   330,1916   690,3024   1317,2783   1359,2791   1361,2792   1500,1569   1682,3029   2023,2275   2786,3022   2795,3021

X(3023) = reflection of X(3027) in X(1)
X(3023) = X(99)-of-Mandart-incircle triangle
X(3023) = homothetic center of intangents triangle and reflection of extangents triangle in X(99)

### X(3024) = 8th STEVANOVIC POINT

Trilinears    a(b + c - a)(b - c)2(b2 + c2 - a2 + bc)2 : :

The circumcircle of the incentral triangle intersects the incircle in 2 points, X(11) and X(3024). (Randy Hutson, August 29, 2018)

X(3024) lies on the incircle and these lines: 1,3028   11,125   12,113   33,1112   35,1511   56,74   65,1354   146,388   690,3023   1317,2771   1359,2778   1361,2779   1469,2781   1682,3031   2774,3022   2836,3021

X(3024) = reflection of X(3028) in X(1)
X(3024) = X(1029)-Ceva conjugate of X(650)
X(3024) = incentral isogonal conjugate of X(523)
X(3024) = X(930)-of-intouch-triangle
X(3024) = X(110)-of-Mandart-incircle triangle
X(3024) = homothetic center of intangents triangle and reflection of extangents triangle in X(110)

### X(3025) = 9th STEVANOVIC POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(b - c)2(b2 + c2 - a2 - bc)2

X(3025) lies on the incircle and these lines: 11,513   36,1464   56,953   517,1317   840,901   1319,1361

X(3025) = X(476)-of-intouch-triangle
X(3024) = reflection of X(11) in line X(1)X(3)

### X(3026) = 10th STEVANOVIC POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c - a)(b - c)2(a2 + ab + ac + 2bc)2
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3026) lies on the incircle and these lines: 150,388   274,1682   1111,1357   1365,1565

Antipodal Pairs on Circles

In response to Stevanovic's findings (X(3000) to X(3006)), Peter Moses noted (Hyacinthos, 12/9/2004) a method for mapping a pair of antipodal points on one circle to an antipodal pair on another circle. The method depends on centers of similitude:

Suppose O1 and O2 are circles, that P is on O1 and that P' is the antipode of P on O1. Let U be the internal center of similitude (insimilicenter) of O1 and O2, and V the exsimilicenter. Define Q = PU ∩P' V and Q' = PV ∩P' U. Then on O2, point Q' is the antipode of Q. Moreover, the lines PP' and QQ' are parallel.

The method follows from introductory material on centers of similitude (e.g., H. S. M. Coxeter, Introduction to Geometry, 2nd edition, page 70, which cites Nathan Altshiller-Court, College Geometry, 2nd edition, page 184). If you have The Geometer's Sketchpad, you can view Antipodal Pairs.

In the following examples, suppose P = p : q : r (trilinears).

Example 1. O1 = circumcircle and O2 = incircle. In this case, insimilicenter(O1, O2) = X(55), and exsimilicenter(O1, O2) = X(56). The antipodal points on O2 given by the Moses mapping are

Q = ((b - c)2p + a(bq + cr))(b + c - a) : :

Q' = ((b + c)2p + a(bq + cr))/(b + c - a) : : .

Example 2. O1 = circumcircle and O2 = Apollonius circle, with center X(970). Here, insimilicenter(O1, O2) = X(573), and exsimilicenter(O1, O2) = X(386). A point on O2 is

Q' = (a + b)(a + c)(b + c)2p - a(bc + ca + ab + b2 + c2)(bq + cr) : : .

Example 3. O1 = circumcircle and O2 = Spieker circle. In this case, insimilicenter(O1, O2) = X(958), and exsimilicenter(O1, O2) = X(1376).

Q = (a - b - c)[((b + c)(b - c)2 + a(b2 + c2))p + (ab + ac + 2bc + a2)(bq + cr)] : :

Q' = (b3 + c3 - ab2 - ac2 - bc2 - b2c)p + (ab + ac - 2bc - a2)(bq + cr) : :

Example 4. O1 = incircle and O2 = Spieker circle. In this case, insimilicenter(O1, O2) = X(2), and exsimilicenter(O1, O2) = X(8).

Q = (bp + cq)/a : (cr + ap)/b : (ap + bq)/c = complement of P

Q' = 2abc(b + c)p - bc(a - b - c)(bq + cr) : :

Example 5. O1 = circumcircle and O2 = sine-triple-angle circle. In this case, insimilicenter(O1, O2) = X(1147), and exsimilicenter(O1, O2) = X(184). A point on O2 is

Q = a[a(a2 - b2)(a2 - c2)p - a2(b2 -a2 + c2)(bq + cr)] : :

### X(3027) = INCIRCLE-ANTIPODE OF X(3023)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c)2(a2 - bc)2/(b + c - a)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3027) lies on the incircle and these lines: 1,2782   11,114   12,115   56,99   65,1356   147,497   148,388   192,1916   226,1365   350,1281   553,1357   690,3028   950,2784   1359,2798   2023,2276   2788,3021

X(3027) = reflection of X(3023) in X(1)
X(3027) = crosssum of X(55) and X(2311)
X(3027) = X(98)-of-Mandart-incircle triangle
X(3027) = homothetic center of intangents triangle and reflection of extangents triangle in X(98)

### X(3028) = INCIRCLE-ANTIPODE OF X(3024)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c)2(1 - 2 cos A)2/(b + c - a)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3028) lies on the incircle and these lines: 1,3024   11,113   12,125   34,1112   36,1464   56,110   57,2948   65,1365   146,497   690,3027   1359,2850   1367,1439   1469,2854   1870,1986   2646,2779   2772,3022   2775,3021

X(3028) = reflection of X(3024) in X(1)
X(3028) = crosssum of X(55) and X(2341)
X(3028) = X(1141)-of-intouch-triangle
X(3028) = X(74)-of-Mandart-incircle-triangle
X(3028) = homothetic center of intangents triangle and reflection of extangents triangle in X(74)

### X(3029) = INTERSECTION X(10)X(115)∩X(98)X(573)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c)2(a3 + a2c - ab2 - b3)(a3 + a2b - ac2 - c3)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3029) lies on the Apollonius circle and these lines: 10,115   98,573   99,386   114,2051   181,3027   690,2782   970,2782   1682,3023   2786,3033   2787,3032   2795,3034   2796,3030

### X(3030) = INTERSECTION X(10)X(11)∩X(43)X(57)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[(2bc - ab - ac)2 - (b2 - c2)2]

Let A' be the inverse-in-excircles-radical-circle of the midpoint of BC, and define B and C' cyclically. Triangle A'B'C' is inscribed in the Apollonius circle and homothetic to the excentral triangle. X(3030) = X(107) of A'B'C'. (Randy Hutson, August 29, 2018))

X(3030) lies on the Apollonius circle and these lines: 10,11   43,57   106,386   373,2177   1293,2291   2796,3029   2832,3034   2842,3031

X(3030) = reflection of X(1357) in X(1054)
X(3030) = X(110)-of-Apollonius-triangle

### X(3031) = INTERSECTION X(10)X(125)∩X(74)X(573)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)2(a3 + b3 - c3 + a2c - b2c - ac2)(a3 - b3 + c3 + a2b - bc2 - ab2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3031) lies on the Apollonius circle and these lines: 10,125   43,2948   74,573   110,386   113,2051   690,3029   1017,2092   2774,3033   2836,3034   2842,3030

### X(3032) = INTERSECTION X(10)X(11)∩X(100)X(386)

Trilinears    (b3 + b2c - ba2 + abc - a2c - ac2)(c3 + c2b - ca2 + abc - a2b - ab2) : :

Let A' be the inverse-in-excircles-radical-circle of the midpoint of BC, and define B and C' cyclically. Triangle A'B'C' is inscribed in the Apollonius circle and homothetic to the excentral triangle. X(3032) = X(74) of A'B'C'. (Randy Hutson, August 29, 2018)

X(3032) lies on the Apollonius circle and these lines: 10,11   100,386   104,573   119,2051   214,1015   528,3034   952,970   1695,1768   2787,3029

### X(3033) = INTERSECTION X(10)X(116)∩X(43)X(57)

Trilinears    a[(a2 - bc)2(b + c)2 - (b3 - c3)2] : :

Let A' be the inverse-in-excircles-radical-circle of the midpoint of BC, and define B and C' cyclically. Triangle A'B'C' is inscribed in the Apollonius circle and homothetic to the excentral triangle. X(3032) = X(98) of A'B'C'. (Randy Hutson, August 29, 2018)

X(3033) lies on the Apollonius circle and these lines: 10,116   43,57   101,386   103,573   118,2051   440,2968   1490,2808   2774,3031   2786,3029

### X(3034) = INTERSECTION X(10)X(116)∩X(105)X(386)

Trilinears    (b3 - b2c + ba2 - abc + a2c - ac2)(c3 - c2b + ca2 - abc + a2b - ab2) : :

Let A' be the inverse-in-excircles-radical-circle of the midpoint of BC, and define B and C' cyclically. Triangle A'B'C' is inscribed in the Apollonius circle and homothetic to the excentral triangle. X(3030) = X(112) of A'B'C'. (Randy Hutson, August 29, 2018)

X(3034) lies on the Apollonius circle and these lines: 10,116   105,386   528,3032   2795,3029   2832,3030   2836,3031

### X(3035) = COMPLEMENT OF X(11)

Trilinears    bc[(a - b + c)(a - c)2 + (a + b - c)(a - b)2] : :
X(3035) = 3*X(2) + X(100)

Let P = X(100) and G=X(2); let GA be the centroid of the triangle BCP. Define GB and GC cyclically. Then G, GA, GB, GC are the vertices of a quadrilateral that is homothetic to the cyclic quadrilateral having vertices A, B, C, P. The center of homothety is X(3035). Moreover, X(3035) is the centroid of both quadrilaterals and is the Feuerbach point of the medial triangle. (Randy Hutson, 9/23/2011)

X(3035) lies on the Spieker circle and these lines: 1,1145   2,11   3,119   8,1317   9,1768   10,140   12,404   36,529   40,1537   80,1698   104,631   153,2551   230,1575   405,2932   468,1861   474,498   516,1638   549,993   620,2787   632,1484   676,2804   899,1818   908,1155   960,2800   1125,1387   1532,2077   2801,3041   2826,3039   2827,3038

X(3035) = midpoint of X(i) and X(j) for these (i,j): (1,1145), (3,119), (8,1317), (10,214), (11,100), (40,1537), (908,1155), (1532,2077)
X(3035) = reflection of X(i) in X(j) for these (i,j): (1387,1125), (3036,10)
X(3035) = complement of X(11)
X(3035) = X(885)-Ceva conjugate of X(518)
X(3035) = centroid of ABCX(100)
X(3035) = Kosnita(X(100),X(2)) point

### X(3036) = COMPLEMENT OF X(1317)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[(a - 2b + c)2/(a - b + c) + (a + b - 2c)2/(a + b - c)]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3036) lies on the Spieker circle and these lines: 2,1217   8,11   9,80   10,140   104,1376   153,2550   355,1158   519,1387   960,2802   1706,1768

X(3036) = midpoint of X(i) and X(j) for these (i,j): (8,11), (80,1145)
X(3036) = reflection of X(3035) in X(10)
X(3036) = complement of X(1317)

### X(3037) = COMPLEMENT OF X(1356)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[b4(a2 - c2)2/(a - b + c) + c4(a2 - b2)2/(a + b - c)]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3037) lies on the Spieker circle and these lines: 2, 1356   741,958

X(3037) = complement of X(1356)

### X(3038) = COMPLEMENT OF X(1357)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[b2(a - c)2/(a - b + c) + c2(a - b)2/(a + b - c)]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3038) lies on the Spieker circle and these lines: 2,1357   9,1054   106,958   121,124   960,2802   1293,1376   2810,3041   2815,3042   2827,3035   2832,3039

X(3038) = complement of X(1357)

### X(3039) = COMPLEMENT OF X(1358)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[(a - c)2/(a - b + c) + (a - b)2/(a + b - c)]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3039) lies on the Spieker circle and these lines: 2,1358   8,3021   9,80   960,2809   1292,1376   2814,3042   2826,3035   2832,3038   2835,3040

X(3039) = midpoint of X(8) and X(3021)
X(3039) = complement of X(1358)
X(3039) = crosssum of X(6) and X(1357)

### X(3040) = COMPLEMENT OF X(1361)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[b2(1 - cos A - cos C)2/(a - b + c) + c2(1 - cos A - cos B)2/(a + b - c)]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3040) lies on the Spieker circle and these lines: 2,1361   8,1364   10,2818   102,1376   109,958   117,2886   124,1329   151,2550   928,3041   956,1795   960,2800   2835,3039

X(3040) = midpoint of X(8) and X(1364)
X(3040) = reflection of X(3042) in X(10)
X(3040) = complement of X(1361)

### X(3041) = COMPLEMENT OF X(1362)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[b2(a2 - ab - bc + c2)2/(a - b + c) + c2(a2 - ac - bc + b2)2/(a + b - c)]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3041) lies on the Spieker circle and these lines: 2,1362   8,3022   9,1282   10,2808   101,958   103,1376   118,124   150,2551   152,2550   928,3040   960,2809   2801,3035

X(3041) = midpoint of X(8) and X(3022)
X(3041) = complement of X(1362)

### X(3042) = COMPLEMENT OF X(1364)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[(a - b + c)(a - c)2cos2B + (a + b - c)(a - b)2cos2C]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3042) lies on the Spieker circle and these lines: 2,1364   8,1361   10,2818   72,1845   102,958   109,1376   117,1329   118,124   151,2551   474,1795   960,2817   2814,3039   2815,3039

X(3042) = midpoint of X(i) and X(j) for these (i,j): (8,1361), (72,1845)
X(3042) = reflection of X(3040) in X(10)
X(3042) = complement of X(1364)

### X(3043) = INTERSECTION X(4)X(110)∩X(74)X(184)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - 4 cos2A)2 sec A
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3043) lies on the sine-triple-angle circle and these lines: 4,110   49,3047   54,125   74,184   186,323   215,3028   378,399   542,3044   1614,2777   1993,2931   2477,3024   2771,3045   2772,3046   2780,3048

X(3043) = reflection of X(3047) in X(49)

### X(3044) = INTERSECTION X(54)X(114)∩X(99)X(184)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a4 + b4 - a2b2 - a2c2)(a4 + c4 - a2b2 - a2c2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3044) lies on the sine-triple-angle circle and these lines: 49,2782   54,114   98,1147   99,184   215,3023   542,3043   543,3048   690,3047   2477,3027   2786,3046

### X(3045) = INTERSECTION X(54)X(119)∩X(100)X(184)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a3 + b3 - a2b - ab2 - ac2 + abc)(a3 + c3 - a2c - ac2 - ab2 + abc)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3045) lies on the sine-triple-angle circle and these lines: 11,110   49,952   54,119   100,184   2771,3043   2805,3048

### X(3046) = INTERSECTION X(54)X(118)∩X(101)X(184)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(a3 + b3 - c3 - a2b - ab2 + bc2)(a3 - b3 + c3 - a2c - ac2 + b2c)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3046) lies on the sine-triple-angle circle and these lines: 49,2808   54,118   101,184   103,1147   215,3022   2772,3043   2774,3047   2786,3044   2813,3048

### X(3047) = INTERSECTION X(54)X(113)∩X(2)X(98)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(a4 + b4 - c4 - 2a2b2 + b2c2)(a4 - b4 + c4 - 2a2c2 + b2c2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3047) lies on the sine-triple-angle circle and these lines: 2,98   49,3043   54,113   74,1147   156,265   193,1177   206,895   215,3024   690,3044   1112,1994   2477,3028   2774,3046   2854,3048

X(3047) = reflection of X(3043) in X(49)

### X(3048) = INTERSECTION X(110)X(126)∩X(111)X(184)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(a4 + b4 - c4 - 4a2b2 + 3b2c2)(a4 - b4 + c4 - 4a2c2 + 3b2c2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3048) lies on the sine-triple-angle circle and these lines: 110,126   111,184   1147,1296   2780,3043   2805,3045   2813,3046   2854,3047

### X(3049) = INTERSECTION X(6)X(523)∩X(520)X(647)

Trilinears    a2[b sin(A - B) - c sin(A - C)] : :
Trilinears    a3(b2 - c2)(b2 + c2 - a2) : :      (M. Iliev, 5/13/07)

X(3049) lies on these lines: 6,523   112,2713   250,2715   421,2501   512,1692   520,647   669,688   924,2485   1510,2492

X(3049) = midpoint of X(i) and X(j) for these (i,j): (6,3050), (2451,3288)
X(3049) = reflection of X(2451) in X(6)
X(3049) = isogonal conjugate of X(6331)
X(3049) = X(i)-Ceva conjugate of X(j) for these (i,j): (2623,512), (2715,237)
X(3049) = cevapoint of X(647) and X(2524)
X(3049) = crosspoint of X(i) and X(j) for these (i,j): (6,1576), (512,647)
X(3049) = crosssum of X(i) and X(j) for these (i,j): (2,950), (4,2489), (99,648), (427,2501), (647,1899)
X(3049) = crossdifference of every pair of points on line X(4)X(69)
X(3049) = isogonal conjugate of isotomic conjugate of X(647)
X(3049) = perspector of hyperbola {{A,B,C,X(3),X(25)}}
X(3049) = barycentric product of PU(109)

### X(3050) = INTERSECTION X(6)X(523)∩X(50)X(647)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 - c2)(a4 - a2b2 - a2c2 - b2c2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3050) lies on these lines: 6,523   50,647   512,1691   520,2506   669,2513

X(3050) = midpoint of X(3049) and X(3288)
X(3050) = reflection of X(6) in X(3049)
X(3050) = X(1576)-Ceva conjugate of X(6)
X(3050) = crosspoint of X(i) and X(j) for these (i,j): (83,110), (112,275)
X(3050) = crosssum of X(i) and X(j) for these (i,j): (39,523), (216,525)
X(3050) = crossdifference of every pair of points on line X(5)X(141) (the complement of the Brocard axis)

### X(3051) = INTERSECTION X(2)X(6)∩X(32)X(184)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(b2 + c2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let L be the isogonal conjugate of the isotomic conjugate of the Brocard axis (i.e., line X(32)X(184)). Let M be the isotomic conjugate of the isogonal conjugate of the Brocard axis (i.e., line X(2)X(69)). Then X(3051) = LnM. (Randy Hutson, March 21, 2019)

X(3051) lies on these lines: 2,6   25,263   31,1911   32,184   42,1197   51,1196   99,703   110,251   213,2308   321,2235   511,1194   669,881   1078,1207   1180,2979   1627,1691

X(3051) = isogonal conjugate of X(308)
X(3051) = X(i)-Ceva conjugate of X(j) for these (i,j): (6,39), (110,669), (694,237), (1576,2531)
X(3051) = X(2531)-cross conjugate of X(1576)
X(3051) = crosspoint of X(i) and X(j) for these (i,j): (6,32), (39,1843)
X(3051) = crosssum of X(i) and X(j) for these (i,j): (2,76), (83,1799)

X(3051) = crossdifference of every pair of points on line X(316)X(512) (the anticomplement of the Lemoine axis)
X(3051) = Danneels point of X(6)
X(3051) = X(92)-isoconjugate of X(1799)

### X(3052) = INTERSECTION X(31)X(42)∩X(32)X(220)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(3a - b - c)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let A'B'C' be the tangential triangle of ABC, and let L be the line through X(1) parallel to BC. Let A'' = L∩B'C', and define B'' and C'' cyclically. Let A* = B'B''∩C'C'', and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X*(3445), and the lines A'A*, B'B*, C'C* concur in X(3052). (Angel Montesdeoca, April 29, 2016)

X(3052) lies on these lines: 3,595   6,31   31,42   32,220   40,1104   44,200   45,612   56,1149   57,1279   101,1384   109,1407   154,2352   171,1001   221,1464   222,2078   238,1376   518,1707   614,1155   692,1397   1260,2220   1333,2256   1460,1486   1836,3011   2176,2223   2192,2342

X(3052) = isogonal conjugate of X(4373)
X(3052) = crossdifference of every pair of points on line X(514)X(4521) (the complement of the Gergonne line)
X(3052) = X(56)-Ceva conjugate of X(6)
X(3052) = crosspoint of X(i) and X(j) for these (i,j): (109,1252), (1420,1743)
X(3052) = crosssum of X(522) and X(1086)

### X(3053) = INTERSECTION X(3)X(6)∩X(4)X(230)

Trilinears    a(3a2 - b2 - c2) : :
Trilinears    a(a2 - SA) : :
Trilinears    sin A - 2 cos A tan ω : :
Trilinears    2 cos A - sin A cot ω : :
Trilinears    a - 4R cos A tan ω : :
Trilinears    a (cot A - cot B cot C) : :

Let Γ1 be the circumcircle, Γ2 the 2nd Lemoine circle, and Γ3 the circle {{X(371),X(372),PU(1),PU(39)}} (which has center X(32)). The three circles intersect in two points, at which let L and L' be the lines tangent to Γ1. Then X(3035) = L∩L'. (Randy Hutson, January 29, 2015)

X(3053) lies on these lines: 3,6   4,230   22,1184   24,112   25,1611   35,609   56,1914   64,248   76,1003   115,382   140,2548   154,237   218,2251   550,2549   988,1100   999,2241   1385,1572   1498,1971   2176,2223

X(3053) = isogonal conjugate of X(2996)
X(3053) = inverse-in-circumcircle of X(1692)
X(3053) = X(25)-Ceva conjugate of X(6)
X(3053) = crosspoint of X(i) and X(j) for these (i,j): (112,249), (193,459)
X(3053) = crosssum of X(115) and X(525)
X(3053) = radical center of Lucas(-cot ω) circles
X(3053) = vertex, other than X(6), of the hyperbola {{X(6),PU(1),PU(2)}}
X(3053) = intersection of diagonals of trapezoid PU(2)PU(39)
X(3053) = insimilicenter of circles with diameters X(371)X(372) and X(1151)X(1152)
X(3053) = trilinear pole wrt tangential triangle of orthic axis X(3053) = crossdifference of every pair of points on line X(523)X(4885) (complement of orthic axis) X(3053) = {X(1687),X(1688)}-harmonic conjugate of X(5171)
X(3053) = insimilicenter of circle centered at X(1151) through X(372) and circle centered at X(1152) through X(371); the exsimilicenter is X(1350)
X(3053) = insimilicenter of circle centered at X(371) through X(1152) and circle centered at X(372) through X(1151); the exsimilicenter is X(1351)

### X(3054) = CENTER OF EVANS CONIC

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(4a4 + 3b4 + 3c4 - 5a2b2 - 5a2c2 - 6b2c2)
Barycentrics   12σ - a2cot ω : 12σ - b2cot ω : 12σ - c2cot ω

See X(13) for a discussion of the Evans conic. If you have The Geometer's Sketchpad, you can view X(3054).

X(3054) and X(3055) were contributed by Peter J. C. Moses, Jan 14, 2005.

X(3054) lies on these lines: 2,6   5,187   39,632   53,468   111,930   115,549   140,574   216,2493   626,2031   1384,1656

X(3054) = midpoint of X(590) and X(615)

### X(3055) = {X(2),X(6)}-HARMONIC CONJUGATE OF X(3054)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(2a4 + 3b4 + 3c4 - 7a2b2 - 7a2c2 - 6b2c2)
Barycentrics   12σ + a2cot ω : 12σ + b2cot ω : 12σ + c2cot ω

X(3055) lies on these lines: 2,6   5,574   32,632   115,547   140,187   570,2493   1384,2548

### X(3056) = INTERSECTION OF LINES X(1)X(256) AND X(6)X(31)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(b2 + c2 - bc)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3056) lies on these lines: 1,256   3,613   6,31   11,141   33,1843   35,182   37,263   51,612   56,1350   69,350   87,291   144,145   210,391   573,2223   611,1351

X(3056) = reflection of X(1469) in X(1)
X(3056) = crosspoint of X(i) and X(j) for these (i,j): (1,2319), (284,314), (982,3061)
X(3056) = crosssum of X(i) and X(j) for these (i,j): (1,1423), (226,1402)
X(3056) = X(6)-of-Mandart-incircle-triangle
X(3056) = homothetic center of intangents triangle and reflection of extangents triangle in X(6)

### X(3057) = INTERSECTION OF LINES X(1)X(3) AND X(10)X(11)

Trilinears    (b + c - a)(b2 + c2 - 2bc + ab + ac) : :
Trilinears    sin2(B/2) + sin2(C/2) : :
Trilinears    2 - cos B - cos C : :
Trilinears    b(tan B/2) + c(tan C/2) : :
X(3057) = (R -r)*X(1) + r*X(3)

Let Oa be the circle centered at the A-excenter and passing trough the A-intouch point. Define Ob and Oc cyclically. The radical center of Oa, Ob, Oc is X(3057). (Randy Hutson, February 10, 2016)

X(3057) divides segment X(1)X(3) in the ratio r/(R-r). (Randy Hutson, February 10, 2016)

X(3057) is the radical center of circles {Oa}, {Ob}, {Oc} used in the construction of the Ursa-minor and Ursa-major triangles; see preamble before X(17603). (Randy Hutson, June 27, 2018)

Let A'B'C' be the excentral triangle. X(3057) is the radical center of the 2nd Droz-Farny circles of triangles A'BC, B'CA, C'AB. (Randy Hutson, June 27, 2018)

X(3057) lies on these lines: 1,3   4,1000   8,210   10,11   12,946   19,2256   21,643   33,1829   34,1902   37,1953   45,374   72,519   140,1387   144,145   190,1222   200,2136   219,2264   220,2082   227,1457   278,1888   355,1479   388,962   496,1737   595,2361   614,1616   1212,1334   1317,1364   2809,3021

X(3057) = reflection of X(i) in X(j) for these (i,j): (8,960), (65,1)
X(3057) = isogonal conjugate of X(1476)
X(3057) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,1201), (190,650), (1293,513)
X(3057) = cevapoint of X(1) and X(2943)
X(3057) = crosspoint of X(1) and X(8)
X(3057) = crosssum of X(i) and X(j) for these (i,j): (56,269), (1,56), (55,1743)

X(3057) = {X(1),X(40)}-harmonic conjugate of X(56)
X(3057) = inverse-in-Feuerbach-hyperbola of X(10)
X(3057) = X(1) of Mandart-incircle triangle
X(3057) = homothetic center of intangents triangle and reflection of extangents triangle in X(1)
X(3057) = bicentric sum of PU(59)
X(3057) = PU(59)-harmonic conjugate of X(650)
X(3057) = {X(1),X(3)}-harmonic conjugate of X(1319)
X(3057) = {X(1),X(65)}-harmonic conjugate of X(354)
X(3057) = X(20) of intouch triangle
X(3057) = X(8) of X(1)-Brocard triangle
X(3057) = perspector of Mandart-incircle triangle and Hutson intouch triangle
X(3057) = outer-Garcia-to-inner-Garcia similarity image of X(11)
X(3057) = X(1885)-of-excentral-triangle

### X(3058) = INTERSECTION OF LINES X(1)X(30) AND X(2)X(11)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c - a)(2a2 + b2 + c2 - 2bc)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(3058) = (R/r)*X(1) - X(2) + X(3)

X(3058) lies on these lines: 1,30   2,11   12,381   72,519   376,1250   496,549   516,553   529,2098   541,3028   542,3024   543,3023   544,3022   551,2646   595,3017   597,2330

X(3058) = inverse-in-Feuerbach-hyperbola of X(3826)
X(3058) = X(2)-of-Mandart-incircle-triangle
X(3058) = homothetic center of intangents triangle and reflection of extangents triangle in X(2)

### X(3059) = INTERSECTION OF LINES X(7)X(8) AND X(9)X(55)

Trilinears    (b + c - a)2(b2 + c2 - 2bc -ab - ac) : :

Let A'B'C' be the intouch triangle of the extangents triangle, if ABC is acute. Then A'B'C' is perspective to the extouch triangle at X(3059). (Randy Hutson, December 2, 2017)

X(3059) lies on these lines: 7,8   9,55   37,2340   72,516   78,1001   142,354   971,2951

X(3059) = reflection of X(i) in X(j) for these (i,j): (65,2550), (390,960)
X(3059) = cevapoint of X(1) and X(2942)
X(3059) = crosspoint of X(8) and X(200)
X(3059) = crosssum of X(i) and X(j) for these (i,j): (56,269), (57,1617)
X(3059) = complement of X(30628)

### X(3060) = INTERSECTION OF LINES X(2)X(51) AND X(4)X(52)

Trilinears    a(b2c2 + c2a2 + a2b2 - b4 - c4) : :
Trilinears    sin A (sin 2B + sin 2C) - sin B sin C : :

X(3060) is the second of three Spanish Points developed by Antreas P. Hatzipolakis and Javier Garcia Capitan in 2009; see X(3567).

X(3060) = centroid of the intersections, other than A, B, C, of circles {{X(4),B,C}}, {{X(4),C,A}}, {{X(4),A,B}} and lines BC, CA, AB. (Randy Hutson, March 25, 2016)

Let A' be the trilinear pole of the perpendicular bisector of BC, and define B' and C' cyclically. A'B'C' is also the anticomplement of the anticomplement of the midheight triangle. X(3060) = X(2)-of-A'B'C'. (Randy Hutson, January 29, 2018)

X(3060) lies on these lines: 2,51   3,143   4,52   6,22   20,389   23,184   25,110   26,54   143,1173   156,195   371,588   372,589   394,1995   648,1629   1853,2781   1992,2393

X(3060) = centroid of reflection triangle
X(3060) = {X(2),X(51)}-harmonic conjugate of X(5640)
X(3060) = X(1699)-of-orthic-triangle, if ABC is acute; see X(3567)
X(3060) = homothetic center of orthic-of-orthic triangle and circumorthic-of-circumorthic triangle
X(3060) = homothetic center of orthocentroidal triangle and X(2)-adjunct anti-altimedial triangle
X(3060) = X(3448)-of-orthocentroidal-triangle
X(3060) = exsimilicenter of the circumcircle and the nine-point circle of the orthic triangle. (Peter Moses, July 1, 2009)

### X(3061) = INTERSECTION OF LINES X(1)X(6) AND X(2)X(257)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b2 + c2 - bc)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3061) lies on these lines: 1,6   2,257   10,262   21,2344   39,986   63,1429   78,2082   169,997   304,1921   330,335

X(3061) = isogonal conjugate of X(7132)
X(3061) = complement of X(3212)
X(3061) = X(3056)-cross conjugate of X(982)
X(3061) = crosssum of X(6) and X(1403)

### X(3062) = ISOGONAL CONJUGATE OF X(165)

Trilinears    1/[tan(B/2) + tan(C/2) - tan(A/2)] :
Barycentrics    a/[tan(B/2) + tan(C/2) - tan(A/2)] : :

If you have The Geometer's Sketchpad, you can view X(3062).

X(3062) lies on these lines: 1,971   7,1699   8,144   9,165   80,2093   269,2310   294,1721   515,1000   943,1490   1156,1445   1320,2801   2335,2947

X(3062) = reflection of X(i) in X(j) for these (i,j): (1768,1156), (2951,9)
X(3062) = isogonal conjugate of X(165)
X(3062) = cevapoint of X(513) and X(2310)
X(3062) = X(57)-cross conjugate of X(1)
X(3062) = crosssum of X(55) and X(1615)
X(3062) = perspector of ABC and 2nd antipedal triangle of X(1)
X(3062) = trilinear product of vertices of 6th mixtilinear triangle

### X(3063) = CROSSDIFFERENCE OF X(7) AND X(8)

Trilinears       a2(b - c)(b + c - a) : b2(c - a)(c + a - b) : c2(a - b)(a + b - c)
Barycentrics  a3(b - c)(b + c - a) : b3(c - a)(c + a - b) : c3(a - b)(a + b - c)

X(3063) is the perspector of the triangle ABC and the tangential triangle of the conic that passes through the points A, B, C, X(55), and X(56).

X(3063) lies on these lines: 6,513   59,919   112,2714   521,650   649,854   657,663   661,832   665,1459   667,788   834,2483

X(3063) = X(i)-Ceva conjugate of X(j) for these (i,j): (644,55), (649,667), (692,2175), (919,2223), (1415,31), (2192, 3022), (2423,1960)
X(3063) = isogonal conjugate of X(4554)
X(3063) = cevapoint of X(798) and X(3049)
X(3063) = crosspoint of X(i) and X(j) for these (i,j): (6,692), (31,1415), (55,644), (649,663), (651,1037)
X(3063) = crosssum of X(i) and X(j) for these (i,j): (2,693), (37,2533), (190,664), (497,650), (513,2275)
X(3063) = crossdifference of every pair of points on the line X(7)X(8)
X(3063) = intersection of trilinear polars of X(55) and X(56)
X(3063) = trilinear product of PU(103)

### X(3064) = ISOGONAL CONJUGATE OF X(1813)

Trilinears    (cos B - cos C)/cos A : (cos C - cos A)/cos B : (cos A - cos B)/cos C
Barycentrics   (cos B - cos C) tan A : (cos C - cos A) tan B : (cos A - cos B) tan C

X(3064) lies on radical axis of Mandart circle and excircles radical circle. (Randy Hutson, December 2, 2017)

X(3064) lies on these lines: 19,649   112,2689   225,770   230,231   243,522

X(3064) = isogonal conjugate of X(1813)
X(3064) = X(i)-Ceva conjugate of X(j) for these (i,j): (108,1856), (158,2310), (278,11), (653,4), (1783,1826), (1897,33)
X(3064) = cevapoint of X(661) and X(2501)
X(3064) = X(i)-cross conjugate of X(j) for these (i,j): (661,650), (663,522), (2170,19), (2310,158)
X(3064) = crosspoint of X(i) and X(j) for these (i,j): (4,653), (92,1897), (1172,1783)
X(3064) = crosssum of X(i) and X(j) for these (i,j): (3,652), (48,1459), 905,1214)
X(3064) = crossdifference of every pair of points on the line X(3)X(73)
X(3064) = pole wrt polar circle of trilinear polar of X(664) (line X(2)X(7))
X(3064) = polar conjugate of X(664)
X(3064) = trilinear pole of line X(2310)X(8735)
X(3064) = trilinear product X(4)*X(650)

### X(3065) = ISOGONAL CONJUGATE OF X(484)

Trilinears       f(A,B,C): f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/[1 + 2(cos A - cos B - cos C)]
Barycentrics  af(A,B,C): bf(B,C,A) : cf(C,A,B)

X(3065) lies on the Neuberg cubic and these lines: 1,399   4,1768   8,191   9,1030   11,79   21,214   30,80   202,1251   314,1227   758,1320   1157,3465   1389,2800   2132,3466   2346,2801

X(3065) = reflection of X(i) in X(j) for these (i,j): (79,11), (484,1749)
X(3065) = isogonal conjugate of X(484)
X(3065) = cevapoint of X(i) and X(j) for these (i,j): (654,2310), (1962,2245)
X(3065) = X(36)-cross conjugate of X(1)<
X(3065) = trilinear pole of line X(650)X(1100)
X(3065) = antigonal conjugate of X(79)
X(3065) = X(2914)-of-excentral-triangle

### X(3066) = 2nd LEMOINE HOMOTHETIC CENTER

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 + 3b2 - c2)(a2 - b2 + 3c2)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Suppose X is a point, with isogonal conjugate X- 1. It is well known that the pedal triangle of X is homothetic to the antipedal triangle of X- 1. The Lemoine homothetic center, X(1285), is the homothetic center when X = X(6), and X(3066) is the homothetic center when X = X(2). See also X(1285). (Peter Moses, Dec. 7, 2005)

In general if X = x : y : z (trilinears), then the homothetic center is given by

a(x + y cos C)(x + z cos B) : b(y + z cos A)(y + x cos C): c(z + x cos B)(z + y cos A).

This is a correction for trilinears given in TCCT, p. 188.

X(3066) lies on these lines: 2,1350   3,373   6,110   25,182   51,394   107,458   125,381

X(3066) = barycentric product of vertices of submedial triangle

### X(3067) = HOFSTADTER ELLIPSE INTERSECTION

Trilinears    a/(AB - AC) : b/(BC - BA) : c/(CA -CB)
Barycentrics    a2/(AB - AC) : b2/(BC - BA) : c2/(CA -CB)

The Hofstadter ellipse E(0) is described at X(359). The point other than A, B, and C in which E(0) meets the circumcircle is X(3067). As with X(359) and X(360), this is a transcendental center, with "exposed angles" A, B, C in its coordinates.

### X(3068) = INTERSECTION OF X(2)X(6) AND X(4)X(371)

Trilinears    sin A + sin B sin C : :
Trilinears    (S + a2)/a : :
Trilinears    cos A + sin A + cos B cos C : :
Barycentrics    (sin A)(sin A + sin B sin C) : :
Barycentrics    a2 + S : :
Barycentrics    SB + SC + S : :

X(3068): Let A'B'C' be the outer Vecten triangle. Let A" be the trilinear pole of line B'C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(3068). (Randy Hutson, July 21, 2017)

X(3068) lies on these lines: {1,1336}, {2,6}, {3,1587}, {4,371}, {5,1588}, {8,7969}, {9,5393}, {10,13893}, {11,13895}, {12,13896}, {13,13917}, {14,13916}, {17,22921}, {18,22876}, {19,1659}, {20,1151}, {24,8276}, {25,13884}, {30,6221}, {32,638}, {39,8963}, {40,13912}, {53,19410}, {54,8995}, {55,13887}, {56,18965}, {64,8991}, {68,13909}, {74,8994}, {76,8992}, {79,16148}, {80,8988}, {83,5491}, {84,8987}, {98,8980}, {99,8997}, {100,13922}, {104,13913}, {110,8998}, {112,13923}, {113,19060}, {114,19056}, {115,19109}, {119,19082}, {125,19111}, {127,19115}, {132,19094}, {140,3312}, {148,13657}, {171,606}, {184,18924}, {186,9682}, {214,19078}, {216,1590}, {238,605}, {239,1267}, {262,22720}, {265,13915}, {372,631}, {376,6200}, {381,6199}, {382,23253}, {387,2047}, {388,2067}, {393,493}, {402,13894}, {427,5410}, {468,5411}, {486,3090}, {487,5286}, {488,6423}, {489,5254}, {490,3053}, {494,13900}, {497,2066}, {498,3301}, {499,3299}, {511,22717}, {515,9583}, {516,9616}, {530,13646}, {531,13645}, {532,22918}, {533,22873}, {538,13647}, {542,13640}, {543,13642}, {546,23263}, {548,6455}, {549,6398}, {550,6449}, {571,13439}, {577,1589}, {588,2165}, {618,19074}, {619,19076}, {620,19108}, {629,19070}, {630,19072}, {632,6428}, {637,1504}, {640,7375}, {641,13771}, {642,19105}, {671,13908}, {754,13648}, {894,5391}, {920,1123}, {946,1702}, {958,19014}, {1001,18999}, {1030,21566}, {1033,15210}, {1075,8954}, {1078,18993}, {1124,3086}, {1125,13959}, {1131,3146}, {1132,3590}, {1147,19062}, {1152,3523}, {1181,6807}, {1209,19096}, {1249,3535}, {1297,13918}, {1327,13920}, {1328,13848}, {1335,3085}, {1370,11417}, {1376,19000}, {1378,19843}, {1449,5405}, {1503,7374}, {1511,19052}, {1579,7400}, {1583,8573}, {1586,3087}, {1599,1609}, {1650,19018}, {1656,6417}, {1657,6407}, {1698,13936}, {1703,6684}, {1788,2362}, {1899,18923}, {2043,5335}, {2044,5334}, {2271,21909}, {2482,19058}, {2548,5058}, {2550,5415}, {2883,19088}, {3035,19112}, {3071,3091}, {3074,3077}, {3075,3076}, {3083,3554}, {3084,3553}, {3088,3093}, {3089,3092}, {3096,19012}, {3103,12251}, {3147,10881}, {3183,22838}, {3186,8956}, {3297,14986}, {3317,10195}, {3462,8955}, {3485,16232}, {3522,6409}, {3524,6396}, {3525,5420}, {3526,6418}, {3528,9680}, {3529,6453}, {3530,6450}, {3534,6445}, {3543,6437}, {3545,6565}, {3594,10303}, {3616,7968}, {3624,13971}, {3627,6447}, {3628,6427}, {3634,13947}, {3647,19080}, {3832,23261}, {3855,23275}, {3934,19089}, {4254,16433}, {4296,9634}, {4297,9615}, {4423,13940}, {5020,19005}, {5021,21992}, {5054,6395}, {5055,18510}, {5056,6431}, {5059,6429}, {5067,10577}, {5070,6500}, {5120,16432}, {5124,21567}, {5200,10132}, {5218,5414}, {5305,11313}, {5409,6805}, {5413,6353}, {5432,19037}, {5433,18995}, {5449,19061}, {5461,19057}, {5480,7000}, {5552,19048}, {5597,13890}, {5598,13891}, {5599,19008}, {5600,19010}, {5889,12239}, {5972,19110}, {6036,19055}, {6118,19102}, {6119,19104}, {6193,8909}, {6202,8396}, {6260,19068}, {6292,19092}, {6302,9112}, {6303,9113}, {6392,6462}, {6410,15717}, {6411,10304}, {6412,15692}, {6424,7388}, {6438,15708}, {6446,15693}, {6448,12108}, {6451,8703}, {6452,12100}, {6456,15712}, {6468,15683}, {6480,11001}, {6481,15719}, {6501,13961}, {6502,7288}, {6519,15704}, {6643,10897}, {6669,19073}, {6670,19075}, {6673,19071}, {6674,19069}, {6689,19095}, {6696,19087}, {6699,19059}, {6701,19079}, {6702,19077}, {6704,19091}, {6705,19067}, {6713,19081}, {6720,19114}, {6776,6811}, {6808,10982}, {6813,14853}, {7160,13914}, {7386,11513}, {7392,10961}, {7493,11418}, {7494,11514}, {7615,13660}, {7737,9675}, {7738,11293}, {7815,13938}, {7846,19011}, {8222,19032}, {8223,19034}, {8280,8889}, {8383,19219}, {8882,16032}, {9648,15338}, {9663,15326}, {9681,22644}, {9683,12088}, {9686,13346}, {9690,15681}, {9691,17800}, {9694,11413}, {9695,12082}, {9722,15234}, {9732,21737}, {9780,13973}, {9862,13674}, {10192,17820}, {10198,13965}, {10200,13964}, {10266,13919}, {10527,19050}, {10533,11206}, {10665,11411}, {10819,12383}, {11265,14790}, {11284,13943}, {11294,12323}, {11442,11447}, {11457,11462}, {11485,18585}, {11486,15765}, {11542,18587}, {11543,18586}, {12221,13881}, {12240,15043}, {12317,12375}, {12318,12424}, {12319,12891}, {12320,12960}, {12321,12961}, {12322,12962}, {12324,12964}, {12325,12965}, {12376,20125}, {12864,19086}, {13025,13045}, {13026,13046}, {13089,19098}, {13203,13287}, {13650,13711}, {13662,13720}, {13701,22541}, {13774,15118}, {13821,19100}, {13832,13833}, {15183,19017}, {15722,17851}, {15819,19064}, {18457,18531}, {19051,20304}, {19084,22966}, {19174,19183}, {19420,19436}, {19421,19438}, {22466,22976}, {22555,22960}

X(3068) = isogonal conjugate of X(493)
X(3068) = complement of X(1270)
X(3068) = X(i)-Ceva conjugate of X(j) for these (i,j): (393,3069), (1585,4)
X(3068) = crosspoint of X(2) and X(1131)
X(3068) = crosssum of X(6) and X(1151)
X(3068) = {X(2),X(6)}-harmonic conjugate of X(3069)
X(3068) = X(2)-of-1st-tri-squares-triangle
X(3068) = X(2)-of-1st-tri-squares-central-triangle
X(3068) = orthologic center of these triangles: 1st tri-squares to outer-Vecten
X(3068) = homothetic center of ABC and 3rd tri-squares central triangle

### X(3069) = INTERSECTION OF X(2)X(6) AND X(4)X(372)

Trilinears    sin A - sin B sin C : :
Trilinears    (S - a2)/a : :
Trilinears    cos A - sin A + cos B cos C : :
Barycentrics    (sin A)(sin A - sin B sin C) : :
Barycentrics    a2 - S : :
Barycentrics    SB + SC - S : :

Let A'B'C' be the inner Vecten triangle. Let A" be the trilinear pole of line B'C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(3069). (Randy Hutson, July 21, 2017)

X(3069) lies on these lines: 1,1123   2,6   3,1588   4,372   5,1587   20,1152   32,637   171,605   216,1589   238,606   393,494   489,3053   577,1590   589,2165   894,1267   920,1336   946,1703   1600,1609   3074,3076   3075,3077

X(3069) = isogonal conjugate of X(494)
X(3069) = complement of X(1271)
X(3069) = X(i)-Ceva conjugate of X(j) for these (i,j): (393,3068), (1586,4)
X(3069) = crosspoint of X(2) and X(1132)
X(3069) = crosssum of X(6) and X(1152)
X(3069) = {X(2),X(6)}-harmonic conjugate of X(3068)
X(3069) = X(2)-of-2nd-tri-squares-triangle
X(3069) = X(2)-of-2nd-tri-squares-central-triangle
X(3069) = homothetic center of ABC and 4th tri-squares central triangle

### X(3070) = INTERSECTION OF X(4)X(6) AND X(5)X(372)

Trilinears    sin A + 2 cos B cos C : :
Barycentrics    (sin A)(sin A + 2 cos B cos C) : :

The points X(3070) and X(3071) are on the Evans conic.

X(3070) lies on these lines: 2,490   3,485   4,6   5,372   20,1151   30,371   487,1991   642,2482   1124,1479   1335,1478   1837,2362

X(3070) = complement of X(490)
X(3070) = crosspoint of X(4) and X(485)
X(3070) = crosssum of X(3) and X(371)
X(3070) = {X(5),X(372)}-harmonic conjugate of X(615)

### X(3071) = INTERSECTION OF X(4)X(6) AND X(5)X(371)

Trilinears        sin A - 2 cos B cos C : sin B - 2 cos C cos A : sin C - 2 cos A cos B
Barycentrics   (sin A)(sin A - 2 cos B cos C) : (sin B)(sin B - 2 cos C cos A) : (sin C)(sin C - 2 cos A cos B)

X(3071) lies on these lines: 2,489   3,486   4,6   5,371   11,2067   12,2066   20,1152   30,372   641,2482   1124,1478   1335,1479   1836,2362

X(3071) = complement of X(489)
X(3071) = crosspoint of X(4) and X(486)
X(3071) = crosssum of X(3) and X(372)
X(3071) = {X(5),X(371)}-harmonic conjugate of X(590)

### X(3072) = INTERSECTION OF X(1)X(3) AND X(4)X(31)

Trilinears        sin2A + cos B cos C : sin2B + cos C cos A : sin2C + cos A cos B
Barycentrics   sin3A + sin A cos B cos C : sin3B + sin B cos C cos A : sin3C + sin C cos A cos B

X(3072) lies on these lines: 1,3   2,602   4,31   5,238   10,580   12,2361   20,601   47,1478   58,515   155,1740   181,389   225,2190   255,388   283,1010   411,1064   497,1497   578,1397   595,946   912,1046   944,1468   947,1412   990,1158   1056,1496   1254,1870   2179,2201

### X(3073) = INTERSECTION OF X(3)X(238) AND X(4)X(31)

Trilinears        sin2A - cos B cos C : sin2B - cos C cos A : sin2C - cos A cos B
Barycentrics   sin3A - sin A cos B cos C : sin3B - sin B cos C cos A : sin3C - sin C cos A cos B

X(3073) lies on these lines: 1,90   2,601   3,238   4,31   5,171   11,1399   20,602   21,1064   40,1724   47,1479   55,3074   57,1777   58,946   104,1201   109,1210   255,497   388,1497   515,595   516,580   578,2175   605,1587   606,1588   631,748   651,1066   774,1870   943,2293   1058,1496   1460,1598   1467,2956

### X(3074) = INTERSECTION OF X(1)X(6) AND X(2)X(255)

Trilinears        cos2A + sin B sin C : cos2B + sin C sin A : cos2C + sin A sin B
Barycentrics   (sin A)(cos2A + sin B sin C) : (sin B)(cos2B + sin C sin A) : (sin C)(cos2C + sin A sin B)

X(3074) lies on these lines: 1,6   2,255   3,1745   4,212   5,1936   10,275   12,2361   28,2183   40,1888   47,171   55,3073   73,1006   165,1777   191,1735   201,1870   226,580   283,908   388,602   603,631   1167,1785   1698,1771   1794,1838   2182,2939   3068,3077   3069,3076

### X(3075) = INTERSECTION OF X(1)X(3) AND X(2)X(255)

Trilinears        cos2A - sin B sin C : cos2B - sin C sin A : cos2C - sin A sin B
Barycentrics   (sin A)(cos2A - sin B sin C) : (sin B)(cos2B - sin C sin A) : (sin C)(cos2C - sin A sin B)

X(3075) lies on these lines: 1,3   2,255   4,603   5,1935   11,1399   29,58   47,238   81,1816   109,946   158,1430   222,1745   389,1364   412,1785   750,1406   1393,1870   1699,1777   3068,3076   3069,3077

X(3075) = X(3)-gimel conjugate of X(35)

### X(3076) = INTERSECTION OF X(1)X(606) AND X(47)X(605)

Trilinears        cos2A + sin A : cos2B + sin B : cos2C + sin C
Barycentrics   (sin A)(cos2A + sin A) : (sin B)(cos2B + sin B) : (sin C)(cos2C + sin C)

X(3076) lies on these lines: 1,606   6,255   31,1125   47,605   109,1702   212,372   371,603   601,2066   1587,1936   1588,1935   3068,3075   3069,3074

### X(3077) = INTERSECTION OF X(1)X(605) AND X(47)X(606)

Trilinears        cos2A - sin A : cos2B - sin B : cos2C - sin C
Barycentrics   (sin A)(cos2A - sin A) : (sin B)(cos2B - sin B) : (sin C)(cos2C - sin C)

X(3077) lies on these lines: 1,605   6,255   31,1135   47,606   109,1703   212,371   372,603   602,2067   1587,1935   1588,1936   3068,3074   3069,3075

### X(3078) = DANNEELS POINT OF X(5)

Trilinears   a[cos2(B - C)][b cos(C - A) + c cos (A - B)] : :
Barycentrics   a2[cos2(B - C)][b cos(C - A) + c cos (A - B)] : :

Let A'B'C' be the cevian triangle of a point X = x : y : z (trilinears), let L(A) be the line through A parallel to B'C', and define L(B) and L(C) cyclically. The lines L(A), L(B), L(C) form a triangle homothetic to A'B'C', with homothetic center

ax2(by + cz) : by2(cz + ax) : cz2(ax + by).

We denote this point by D(X) and call it the Danneels point of X. (Eric Danneels, Hyacinthos, Jan. 28, 2005). In addition to the discussion below, see the preamble to X(8012) and the Danneels points beginning at X(8012).

If X lies on the line at infinity, then D(X) = X(2) of cevian triangle of X. (Randy Hutson, Jul 23, 2015)

The formula is simpler in barycentrics: if U = u : v : w, then

D(U) = u2(v + w) : v2(w + u) : w2(u + v).

If X is line the Euler line, then D(X) is on the Euler line. A proof follows. Let a1, b1, c1 be cos A, cos B, cos C, respectively, so that the Euler line is given parametrically as x(t) : y(t) : z (t) by

x = b1c1 + ta1
y = c1a1 + tb1
z = a1b1 + tc1,

where t is an arbitrary function homogenous of degree 0 in a, b, c. Then D(X) is the point U = u : v : w given by

u = ax2(by + cz)
v = by2(cz + ax)
w = cz2(ax + by).

A point P = p : q : r is on the Euler line if

a(b2 - c2)(b2 + c2 - a2)p
+ b(c2 - a2)(c2 + a2 - b2)q
+ c(a2 - b2)(a2 + b2 -c2)r = 0.

It is easy to check that the point U satisfies this equation. (This sort of algebraic proof is more inclusive than a geometric proof, because here, a,b,c are indeterminates or variables. They can, for example, take values that are not sidelengths of a triangle. Here, cos A is defined as (b2 + c2 - a2)/(2bc), so that no dependence on a geometric angle is necessary.)

The appearance of (i,j) in the following list means that D(X(i)) = X(j):
(1,42)   (2,2)   (3,418)   (4,25)   6,3051   (7,57)   (8,200)   (69,394)   (75,321)   (100,55)   (110,84)   (264,324)   (366,367)   (651,222)   (653,196)   (1113,25)   (1114,25)   (1370,455) For an extension of this list, see the preamble to X(8012).

X(3078) lies on this line: (2,3)

X(3078) = X(5)-Ceva conjugate of X(233)

X(3078) = X(288)-isoconjugate of X(2167)

### X(3079) = DANNEELS POINT OF X(20)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a u2(bv + cw), where u : v : w = X(20)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

As a point on the Euler line, X(3079) has Shinagawa coefficients (4F2, -4(E + F)F + S2).

X(3079) lies on these lines: {2, 3}, {154, 1249}, {204, 1394}, {459, 1503}, {1495, 6524}, {1498, 2131}, {5562, 5910}

X(3079) = X(20)-Ceva conjugate of X(1249)

X(3079) = X(1073)-isoconjugate of X(2184)

### X(3080) = DANNEELS POINT OF X(25)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a u2(bv + cw), where u : v : w = X(25)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

As a point on the Euler line, X(3080) has Shinagawa coefficients ((E + F)3F - (E - F)FS2, -(E + F)4 + (E + F)(2E - F)S2).

X(3080) lies on these lines: {2, 3}, {682, 1196}

X(3080) = X(25)-Ceva conjugate of X(1196)

X(3080) =X(326)-isoconjugate of X(683)

### X(3081) = DANNEELS POINT OF X(30)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a u2(bv + cw), where u : v : w = X(30)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

As a point on the Euler line, X(3081) has Shinagawa coefficients (27F2 - S2, -27(E + F)F + 9S2).

X(3081) lies on this line: (2,3)

X(3081) = reflection of X(1650) in X(1651)
X(3081) = X30)-Ceva conjugate of X(3163)

### X(3082) = RADICAL CENTER OF EXTERNAL MALFATTI CIRCLES

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(1 - cos A/2)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

If you have The Geometer's Sketchpad, you can view X(3082).

X(3082) lies on these lines: 8,178   174,176

### X(3083) = INTERSECTION OF LINES X(1)X(2) AND X(37)X(494)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1 + csc A
Trilinears        bc + S : ca + S : ab + S
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3083) lies on these lines: 1,2   11,1591   12,1592   33,1586   34,1585   35,1600   36,1599   38,494   55,1584   56,1583   176,329   394,1124   1038,1590   1040,1589

### X(3084) = INTERSECTION OF LINES X(1)X(2) AND X(37)X(493)

Trilinears    1 - csc A : :
Trilinears    bc - S : ca - S : ab - S

X(3084) lies on these lines: 1,2   11,1592   12,1591   33,1585   34,1586   35,1599   36,1600   37,493   55,1583   56,1584   175,329   394,1335   1038,1589   1040,1590

### X(3085) = INTERSECTION OF LINES X(1)X(2) AND X(4)X(12)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1 + sin B sin C
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3085) lies on these lines: 1,2   3,388   4,12   5,497   7,46   11,1058   20,35   28,197   31,3074   33,3089   34,3088   37,158   40,226   56,631   69,611   100,377   140,999   144,191   171,255   212,3072   227,278   238,1497   346,1089   390,1479   405,2551   42,954   443,1376   496,1656   601,1935   750,1496   756,774   942,1788   944,2646   1124,3069   1335,3068   1588,2066   2241,2548

### X(3086) = INTERSECTION OF LINES X(1)X(2) AND X(4)X(11)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1 - sin B sin C
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3086) lies on these lines: 1,2   3,496   4,11   5,388   7,90   12,1056   20,36   31,3075   33,3088   34,3089   35,390   55,631   57,946   69,613   171,1497   238,255   269,1256   406,1104   443,2886   495,1656   602,1936   603,3073   748,1496   944,1319   1124,3068   1335,3069   1588,2067   2242,2548

X(3086) = crosspoint of X(i) and X(j) for these (i,j): (2,1440), (1123,1336)
X(3086) = crosssum of X(1124) and X(1335)

### X(3087) = INTERSECTION OF LINES X(2)X(95) AND X(4)X(6)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec A + csc B csc C
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3087) lies on these lines: 4,6   19,1877   20,216   29,1778   32,3088

### X(3088) = INTERSECTION OF LINES X(2)X(3) AND X(33)X(3086)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec A + sin B sin C
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(3088) has Shinagawa coefficients (F,E).

X(3088) lies on these lines: 2,3   32,3087   33,3086   34,3085

X(3088) = inverse-in-orthocentroidal-circle of X(3089)

### X(3089) = INTERSECTION OF LINES X(2)X(3) AND X(33)X(3085)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec A - sin B sin C
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(3089) has Shinagawa coefficients (F, -E).

X(3089) lies on these lines: 2,3   33,3085   34,3086   3068,3092   3069,3093

X(3089) = inverse-in-orthocentroidal-circle of X(3088)
X(3089) = anticomplement of X(3546)

### X(3090) = INTERSECTION OF LINES X(2)X(3) AND X(11)X(1058)

Trilinears    sin B sin C + cos(B - C) : :

Trilinears    2 cos A + 3 cos B cos C : :
Trilinears    3 sec A + 2 sec B sec C : :
Barycentrics    a^4 - 4a^2(b^2 + c^2) + 3(b^2 - c^2)^2 : :
Barycentrics    2 S^2 + SB SC : :

As a point on the Euler line, X(3090) has Shinagawa coefficients (2,1).

X(3090) lies on these lines: 2,3   11,1058   12,1056   69,576   76,1007   110,569   155,1199   182,1614   233,1249   373,389   388,499   485,3069   486,3068   497,498   575,1352   590,1588   615,1587   748,3072   750,3073   944,1125   946,1698   1493,2888

X(3090) = reflection of X(i) in X(j) for these (i,j): (3523,3526), (3528,3523)
X(3090) = inverse-in-orthocentroidal-circle of X(631)
X(3090) = complement of X(3523)
X(3090) = homothetic center of orthocentroidal triangle and X(5)-Brocard triangle
X(3090) = homothetic center of anti-Euler triangle and cross-triangle of ABC and Euler triangle
X(3090) = homothetic center of Euler triangle and cross-triangle of ABC and anti-Euler triangle
X(3090) = homothetic center of ABC and cross-triangle of Euler and anti-Euler triangles
X(3090) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,5,3091), (4,5,3545), (4,3544,3091)
X(3090) = homothetic center of X(2)-altimedial and X(3)-anti-altimedial triangles
X(3090) = homothetic center of X(4)-altimedial and X(20)-anti-altimedial triangles

### X(3091) = INTERSECTION OF LINES X(2)X(3) AND X(11)X(153)

Trilinears    cos B cos C + cos(B - C) : :
Barycentrics    S^2 + 2 SB SC : :
Barycentrics    a^4 + 2 a^2 (b^2 + c^2) - 3 (b^2 - c^2)^2 : :

As a point on the Euler line, X(3091) has Shinagawa coefficients (1,2).

Let A'B'C' be the Euler triangle. Let A" be the cevapoint of B' and C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(3091). (Randy Hutson, December 10, 2016)

Let A'B'C' be the Euler triangle. Let A* be the centroid of AB'C', and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(3091). (Randy Hutson, December 10, 2016)

X(3091) lies on these lines: 2,3   7,1210   8,908   10,962   11,153   12,497   68,1173   110,578   114,148   115,147   116,152   118,150   119,149   124,151   125,146   145,355   155,1994   156,567   193,576   194,262   226,938   253,264   324,1093   347,1838   390,1479   485,1132   486,1131   495,1058   496,1056   516,1698   569,1614   637,1271   638,1270   1007,1975   1329,2550   1348,2543   1349,2542   1454,1776   1478,3086   1506,2549   1676,2547   1677,2546   1788,1836   1853,2883   2009,2545   2010,2544   2551,2886   3068,3071   3069,3070

X(3091) = midpoint of X(4) and X(631)
X(3091) = reflection of X(i) in X(j) for these (i,j): (3,632), (631,1656), (1656,5), (3522,631)
X(3091) = inverse-in-orthocentroidal-circle of X(20)
X(3091) = complement of X(3522)
X(3091) = anticomplement of X(631)
X(3091) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,5,3090), (4,5,2), (4,3544,3090)
X(3091) = homothetic center of orthocentroidal triangle and X(4)-Brocard triangle
X(3091) = homothetic center of anticomplementary triangle and cross-triangle of ABC and Euler triangle
X(3091) = homothetic center of Euler triangle and cross-triangle of ABC and Euler triangle
X(3091) = circumcenter of cross-triangle of Euler and anti-Euler triangles
X(3091) = homothetic center of X(2)-altimedial and X(4)-anti-altimedial triangles
X(3091) = homothetic center of X(20)-altimedial and X(20)-anti-altimedial triangles
X(3091) = homothetic center of X(20)-anti-altimedial and orthocentroidal triangles
X(3091) = X(7897)-of-orthic-triangle if ABC is acute
X(3091) = homothetic center of Ehrmann mid-triangle and anti-Euler triangle

### X(3092) = INTERSECTION OF LINES X(2)X(1579) AND X(4)X(6)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + sec A
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3092) lies on these lines: 2,1579   4,6   20,1578   24,1151   25,371   33,1335   34,1124   235,485   372,1593   378,1152   394,637   427,486   1377,1861   3068,3089   3069,3088

### X(3093) = INTERSECTION OF LINES X(2)X(1578) AND X(4)X(6)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A - sec A
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3093) lies on these lines: 2,1578   4,6   20,1579   24,1152   25,372   33,1124   34,1335   235,486   371,1593   378,1151   394,638   427,485   1378,1861   1905,2362   3068,3088   3069,3089

### X(3094) = INTERSECTION OF LINES X(2)X(694) AND X(3)X(6)

Trilinears    sin(A + 2ω) : sin(B + 2ω) : sin(C + 2ω)
Trilinears    a(b4 + c4 + b2c2) : :
Trilinears    e^2 cos(A + ω) - cos(A - ω) : :

Let A2B2C2 be the 2nd Brocard triangle. Let A' = inverse-in-Brocard circle of A, and define B' and C' cyclically. Let A2' = inverse-in-circumcircle of A2, and define B2' and C2' cyclically. Let A'' = B'B2'∩C'C2', and define B'' and C'' cyclically. The lines AA", BB", CC" concur in X(3094). (Randy Hutson, December 26, 2015)

Let A'B'C' be the 3rd Brocard triangle. Then X(3094) is the radical center of the circumcircles of A'BC, B'CA, C'AB. (Randy Hutson, December 26, 2015)

Let D = P(1), E = U(1), F = P(2), G = U(2). Let D' = X(4)-of-EFG, E' = X(4)-of-DFG, F' = X(4)-of-DEG, G' = X(4)-of-DEF. Then D'E'F'G' is a cyclic quadrilateral whose circumcenter is X(3094). (Randy Hutson, December 26, 2015)

X(3094) lies on these lines: 2,694   3,6   22,1915   69,194   76,141   99,737   262,1513   394,2056   538,599   542,1569   726,2321   1180,2979   1194,1613   1352,2549   1469,2276   2275,3056

X(3094) = midpoint of X(69) and X(194)
X(3094) = reflection of X(i) in X(j) for these (i,j): (6,39), (76,141)
X(3094) = isogonal conjugate of X(3407)
X(3094) = inverse-in-Brocard circle of X(1691)
X(3094) = crosssum of X(32) and X(182)
X(3094) = crosspoint of X(76) and X(262)
X(3094) = isotomic conjugate of X(3114)
X(3094) = {X(371),X(372)}-harmonic conjugate of X(3398)
X(3094) = {X(1340),X(1341)}-harmonic conjugate of X(39)
X(3094) = inverse-in-2nd-Brocard-circle of X(32)
X(3094) = X(76)-of-1st-Brocard-triangle
X(3094) = X(6)-of-5th-Brocard-triangle
X(3094) = X(4)-of-X(3)PU(1)
X(3094) = inverse-in-circle-{{X(3102),X(3103),PU(1)}} of X(3)
X(3094) = perspector of 2nd Brocard triangle and unary cofactor triangle of 1st Brocard triangle

### X(3095) = INTERSECTION OF LINES X(3)X(6) AND X(5)X(76)

Trilinears    cos(A + 2ω) : cos(B + 2ω) : cos(C + 2ω)
Trilinears   a(b6 + c6 - a2b4 - a2c4 - 3a2b2c2) : :
Trilinears   e2sin(A + ω) - sin(A - ω) : :

Let U be the pedal triangle of the 1st Brocard point of ABC, and let V be the pedal triangle of the 2nd Brocard point. Let X' = X(39)-of-U and Y' = X(512)-of-U; let X'' = X(39)-of-V and Y'' = X(512)-of-V. Then X(3095) = X'Y'∩X''Y''. (Randy Hutson, September 5 , 2015)

Let X be a point on the 2nd Brocard circle. The locus of X(4) of triangle XPU(1) as X varies is a circle with center X(3095). This circle is the reflection of the 2nd Brocard circle in X(39). See also X(9821). (Randy Hutson, July 20, 2016)

The locus of the orthocenter in a Brocard porism (triangles sharing circumcircle and Brocard inellipse with ABC) is a circle with center X(3095) and segment X(3557)X(3558) as diameter. The circle also passes through X(4) and its antipode X(194). (Randy Hutson, August 29, 2018)

X(3095) lies on these lines: 3,6   4,147   5,76   237,3060   355,730   381,538   726,946   732,1352   1993,2001

X(3095) = midpoint of X(i) and X(j) for these (i,j): (4,194), (3557, 3558)
X(3095) = reflection of X(i) in X(j) for these (i,j): (3,39), (76,5)
X(3095) = isogonal conjugate of X(3406)
X(3095) = inverse-in-Brocard-circle of X(3398)
X(3095) = inverse-in-2nd-Brocard-circle of X(182)
X(3095) = {X(371),X(372)}-harmonic conjugate of X(1691)
X(3095) = harmonic center of 2nd Lemoine circle and the circle {{X(371),X(372),PU(1),PU(39)}}
X(3095) = center of circle {{X(4),X(194),X(3557),X(3558)}}
X(3095) = X(4)-of-X(6)PU(1)
X(3095) = orthologic center of these triangles: Johnson to 1st Neuberg
X(3095) = X(76)-of-Johnson-triangle
X(3095) = inverse-in-circle-{X(3102),X(3103),PU(1)} of X(6)
X(3095) = intersection of lines PU(1) of pedal triangles of PU(1)

### X(3096) = INTERSECTION OF LINES X(2)X(32) AND X(76)X(141)

Trilinears     bc(b4 + c4 + b2c2 + c2a2 + a2b2) : :

X(3096) lies on these lines: 2,32   4,3098   76,141   114,631   211,2979   316,2076

X(3096) = complement of X(7787)
X(3096) = {X(15),X(16)}-harmonic conjugate of X(32)
X(3096) = homothetic center of medial triangle and 5th Brocard triangle
X(3096) = homothetic center of ABC and cross-triangle of ABC and 5th Brocard triangle

### X(3097) = INTERSECTION OF LINES X(1)X(39) AND X(43)X(63)

Trilinears        f(a,b,c)) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a + 2b + 2c)(b2 + c2) - b2c2

X(3097) lies on these lines: 1,39   2,726   10,194   43,63   76,1698   165,511   262,1699   984,1575   1670,1700   1671,1701

X(3097) = reflection of X(1699) in X(262)

### X(3098) = INTERSECTION OF LINES X(3)X(6) AND X(69)X(74)

Trilinears    a(a4 - 2b4 - 2c4 - 2b2c2 + c2a2 + a2b2) : :
Trilinears    sin A - 3 cos A cot ω : :
Trilinears    3 cos A - sin A tan ω : :
Trilinears    2 sin(A + 2ω) - sin(A - 2ω) - sin A : :
X(3098) = 3X(3) - X(6)

X(3098) lies on these lines: 3,6   4,3096   20,1352   22,1495   30,141   35,1469   36,3056   69,74   184,323   186,1974   206,1511   378,1843   399,2916   550,1503   805,842

X(3098) = midpoint of X(i) and X(j) for these (i,j): (3,1350), (20,1352)
X(3098) = reflection of X(i) in X(j) for these (i,j): (182,3), (576,182), (1351,575)
X(3098) = inverse-in-Brocard-circle of X(5092)
X(3098) = {X(371),X(372)}-harmonic conjugate of X(5007)
X(3098) = inverse-in-circle-{{X(371),X(372),PU(1),PU(39)}} of X(5007)
X(3098) = X(7)-of-Trinh-triangle if ABC is acute
X(3098) = Trinh-isotomic conjugate of X(3357)
X(3098) = X(182)-of-circumcevian-triangle-of-X(511)
X(3098) = X(182)-of-5th-Brocard-triangle
X(3098) = 6th-Brocard-to-5th-Brocard similarity image of X(3)

### X(3099) = INTERSECTION OF LINES X(1)X(32) AND X(846)X(902)

Trilinears        f(a,b,c)) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b4 + c4 - a4 - 2a3b - 2a3c + b2c2 + c2a2 + a2b2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3099) lies on these lines: 1,32   10,2896   518,2076   672,1282   726,2959   846,902

### X(3100) = INTERSECTION OF LINES X(1)X(7) AND X(2)X(33)

Trilinears        f(a,b,c)) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b4 + c4 - a4 - bc(b2 + c2 - a2))
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3100) lies on these lines: 1,7   2,33   4,1062   11,858   21,270   22,55   30,1870   36,2071   78,280   165,3100   229,2646   238,2310   376,1060   394,2192   497,1370   518,677   522,663   651,971   655,1807   984,1253   1776,2361

X(3100) = anticomplement of X(1861)
X(3100) = crosssum of X(i) and X(j) for these (i,j): (65,1456), (1409,2223)
X(3100) = crossdifference of every pair of points on the line X(657)X(1400)

### X(3101) = INTERSECTION OF LINES X(2)X(19) AND X(8)X(20)

Trilinears        f(a,b,c)) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a5 + a3bc + (b + c)(a4 - a2bc - a(b3 + c3) - (b - c)(b3 - c3))
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3101) lies on these lines: 2,19   8,20   22,55   27,1441   46,387   57,347   65,81   71,1654   100,1297   306,2897   329,1763   345,1760   573,1726   1172,1214   1305,1952   1370,2550   1442,1790   1770,2960   1869,2475

### X(3102) = INNER VERTEX OF THE BROCARD SQUARE

Trilinears      cos(A + ω - π/4) : :
Trilinears    sin(A + ω + π/4) : :
Trilinears    bc cos2A - a2cos B cos C + ab sin B + ac sin C : :
Trilinears    sin A + cos(A + 2ω) : :
Trilinears    cos A + sin(A + 2ω) : :
Trilinears    cos(A + ω) + sin(A + ω) : :

The 1st Brocard point (trilinears c/b : a/c : b/a) and 2nd Brocard point (b/c : c/a : a/b) are opposing vertices of a square here called the Brocard square. The other two vertices are X(3102) and X(3103). Both are on the Brocard axis X(3)X(6), and X(3102) is the closer to X(3).

X(3102) lies on these lines: 3,6   76,486   194,488   262,485   325,640   2782,3071

X(3102) = reflection of X(3103) in X(39)
X(3102) = inverse-in-2nd-Brocard-circle of X(372)

### X(3103) = OUTER VERTEX OF THE BROCARD SQUARE

Trilinears    cos(A + ω + π/4) : :
Trilinears    sin(A + ω - π/4) : :
Trilinears    bc cos2A - a2cos B cos C - ab sin B - ac sin C : :
Trilinears    sin A - cos(A + 2ω) : :
Trilinears    cos A - sin(A + 2ω) : :
Trilinears    cos(A + ω) - sin(A + ω) : :
X(3103) is the vertex of the Brocard square that is farest from X(3); see X(3102).

X(3103) lies on these lines: 3,6   76,485   194,487   262,486   325,639      2782,3070

X(3103) = reflection of X(3102) in X(39)
X(3103) = inverse-in-2nd-Brocard-circle of X(371)

### X(3104) = INNER VERTEX OF 1st BROCARD EQUILATERAL TRIANGLE

Trilinears        cos(A + ω - π/6) : cos(B + ω - π/6) : cos(C + ω - π/6)
Trilinears        sin(A + ω + π/3) : sin(B + ω + π/3) : sin(C + ω + π/3)
Trilinears        f(a,b,c)) : f(b,c,a) : f(c,a,b), where f(a,b,c) = sqrt(3)(bc cos2A - a2cos B cos C) + ab sin B + ac sin C
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

There are two 60-rotations of the 1st Brocard point about the 2nd; X(3104) is the one closer to X(3).

X(3104) lies on these lines: 3,6   14,76   17,262   194,616      633,2896

X(3104) = reflection of X(i) in X(j) for these (i,j): (3105,39), (3107, 3106)
X(3104) = inverse-in-2nd-Brocard-circle of X(16)

### X(3105) = OUTER VERTEX OF 2nd BROCARD EQUILATERAL TRIANGLE

Trilinears        cos(A + ω + π/6) : cos(B + ω + π/6) : cos(C + ω + π/6)
Trilinears        sin(A + ω - π/3) : sin(B + ω - π/3) : sin(C + ω - π/3)
Trilinears        f(a,b,c)) : f(b,c,a) : f(c,a,b), where f(a,b,c) = sqrt(3)(bc cos2A - a2cos B cos C) - ab sin B - ac sin C
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

There are two 60-rotations of the 1st Brocard point about the 2nd; X(3105) is the one farther from X(3).

X(3105) lies on these lines: 3,6   13,76   18,262   194,617      634,2896

X(3105) = reflection of X(i) in X(j) for these (i,j): (3104,39), (3106, 3107)
X(3105) = inverse-in-2nd-Brocard-circle of X(15)

### X(3106) = CENTER OF 1st BROCARD EQUILATERAL TRIANGLE

Trilinears     cos(A + ω - π/3) : cos(B + ω - π/3) : cos(C + ω - π/3)
Trilinears    sin(A + ω + π/6) : sin(B + ω + π/6) : sin(C + ω + π/6)
Trilinears    bc cos2A - a2cos B cos C + sqrt(3)(ab sin B + ac sin C) : :

X(3106) is the center of the equilateral triangle introduced at X(3104).

X(3106) lies on these lines: 3,6   13,262   14,2782   18,76      194,627

The locus of the 2nd Isogonic Center in a Brocard porism (triangles sharing circumcircle and Brocard inellipse with ABC) is a circle with center X(3106). (Randy Hutson, August 29, 2018)

X(3106) = midpoint of X(3104) and X(3107)
X(3106) = reflection of X(i) in X(j) for these (i,j): (3105,3107), (3107, 39)
X(3106) = inverse-in-2nd-Brocard-circle of X(62)
X(3106) = X(13)-of-X(3)PU(1)
X(3106) = X(14)-of-X(6)PU(1)

### X(3107) = CENTER OF 2nd BROCARD EQUILATERAL TRIANGLE

Trilinears    cos(A + ω + π/3) : cos(B + ω + π/3) : cos(C + ω + π/3)
Trilinears    sin(A + ω - π/6) : sin(B + ω - π/6) : sin(C + ω - π/6)
Trilinears    bc cos2A - a2cos B cos C - sqrt(3)(ab sin B + ac sin C)

X(3107) is the center of the equilateral triangle introduced at X(3105).

The locus of the X(13) in a Brocard porism (triangles sharing circumcircle and Brocard inellipse with ABC) is a circle with center X(3107). (Randy Hutson, August 29, 2018)

X(3107) lies on these lines: 3,6   13,2782   14,262   17,76      194,628

X(3107) = midpoint of X(3105) and X(3106)
X(3107) = reflection of X(i) in X(j) for these (i,j): (3105,3106), (3106, 39)
X(3107) = inverse-in-2nd-Brocard-circle of X(61)
X(3107) = X(14)-of-X(3)PU(1)
X(3107) = X(13)-of-X(6)PU(1)

### X(3108) = DC(X(83))

Trilinears        f(a,b,c)) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(2a2 + b2 + c2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

For an introduction to the DC and CD mappings, see the discussion just before X(2979).

X(3108) lies on these lines: 6,1627   25,1180   39,251   111,1194   305,1239

X(3108) = cevapoint of X(i) and X(j) for these (i,j): (6,39), (1015,2483)
X(3108) = X(3005)-cross conjugate of X(110)

X(3108) = trilinear pole of line X(523)X(2076)
X(3108) = perspector of hyperbola {{A,B,C,X(4),X(476)}}
X(3108) = intersection of trilinear polars of X(4) and X(476)

### X(3109) = ORTHOGONAL PROJECTION OF X(1) ON EULER LINE

Trilinears    bc(2a5 - 2a4(b + c) + 2a3bc + g(a,b,c))/(b + c) : :
Trilinears    (b + c)(b - c)2(b + c - a)2

X(3109) lies on these lines: 1, 523   2,3   11,759   12,2222   110,952   119,1793   163,1146   476,953   643,1145   691,2726   1304,2734

X(3109) = inverse-in-circumcircle of X(859)
X(3109) = inverse-in-polar-circle of X(860)
X(3109) = intersection, other than X(3), of the Euler line and circle O(1,3)
X(3109) = X(110)-of-X(1)-Brocard triangle

### X(3110) = ORTHOGONAL PROJECTION OF X(1) ON BROCARD AXIS

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b4 + c4 + 2a2bc - a(b + c)(b2 + c2))/(b + c)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3110) lies on the circle O(1,3) and these lines: 1, 512   3,6   21,1083   60,249   691,840   741,1015   813,1500   1362,1414

X(3110) = inverse-in-circumcircle of X(3286)
X(3110) = crossdifference of every pair of points on the line X(523)X(2238)
X(3110) = X(99) of X(1)-Brocard triangle

### X(3111) = ORTHOGONAL PROJECTION OF X(2) ON BROCARD AXIS

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(b2 + c2 - a2)(b4 + c4 - 4b2c2) + ab2c2(b4 + c4)

Let A'B'C' be the medial triangle. Let A" be the reflection of X(115) in line B'C', and define B" and C" cyclically. X(3111) is the centroid of (degenerate) triangle A"B"C".

X(3111) lies on these lines: 2, 512   3,6   373,1316

X(3111) = X(99)-of-X(2)-Brocard triangle
X(3111) = Brocard axis intercept, other than X(3), of circle O(2,3)

T and G Mappings
This section presents points X(3112) to X(3118) as representative of points introduced by Quang Tuan Bui in Hyacinthos, August 5, 2006.

Suppose X is a point not on a sideline of ABC. Let

gX = isogonal conjugate of X
tX = isotomic conjugate of X
tgX = isotomic conjugate of gX
gtX = isogonal conjugate of tX
Gt = intersection of lines X(tX) and (gX)(gtX)
Tg = intersection of lines X(gX) and (tX)(tgX)

Then the points A, B, C, gX, tX, Tg, Gt are on a conic. As a circumconic, it is the image under the isogonal conjugate mapping of line X(gtX). It is also the image under the isotomic conjugate mapping of line X(tgX).

If X = x : y : z (trilinears), then

Gt = (b2 - c2)/(x(b2y2 - c2z2)) : (c2 - a2)/(y(c2z2 - a2x2)) : (a2 - b2)/(z(a2x2 - b2y2))

Tg = (b2 - c2)/(x(y2 - z2)) : (c2 - a2)/(y(z2 - x2)) : (a2 - b2)/(z(x2 - y2))

The X(31)-conic passes through X(i) for I = 75, 92, 313, 321, 561, 1441, 1821, 1934, 2995, 2997.
The X(32)-conic passes through X(i) for I = 76, 264, 276, 290, 300, 301, 308, 313, 327, 349, 1502, 2367.
The X(76)-conic passes through X(i) for I = 6, 32, 83, 213, 729, 981, 1918, 1974, 2207, 2281, 2422.

### X(3112) = TgX(31)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/(a2(b2 + c2))

X(3112) lies on these lines: 1,561   31,75   38,799   42,308   83,213   92,1973   689,741   1402,1441

X(3112) = isogonal conjugate of X(1964)
X(3112) = isotomic conjugate of X(38)
X(3112) = cevapoint of X(1) and X(75)
X(3112) = X(i)-cross conjugate of X(j) for these (i,j): (1,82), (661,799), (1215,2), (1580,1821)
X(3112) = crosspoint of X(1) and X(75) wrt the excentral triangle
X(3112) = trilinear pole of line X(798)X(812)

### X(3113) = GtX(31)

Trilinears    (b2 - c2)/(a2(b6 - c6)) : :
Barycentrics    1/(b^2c^2 + b^4 + c^4) : :

X(3113) lies on these lines: 1, 1927   31,561   75,560   313,983   321,2205   870,1441

X(3113) = isogonal conjugate of X(3116)
X(3113) = trilinear pole of line X(1577)X(1924)

### X(3114) = TgX(32)

Trilinears 1/(a3(b4 + c4 + b2c2)) : :

Let A'B'C' be the 3rd Brocard triangle. Let La be the line through A' parallel to BC, and define Lb and Lc cyclically. Let A" = Lb∩Lc, B" = Lc∩La, C" = La∩Lb. The lines AA", BB", CC" concur in X(3114). (Randy Hutson, August 29, 2018)

X(3114) lies on these lines: 6,706   32,76   183,327   264,419   313,983   870,1215

X(3114) = isogonal conjugate of X(3117)
X(3114) = isotomic conjugate of X(3094)
X(3114) = trilinear pole of line X(669)X(804)

### X(3115) = GtX(32)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 - c2)/(a3(b8 - c8))
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3115) lies on these lines: 32,710   76,1501

X(3115) = isogonal conjugate of X(3118)

### X(3116) = ISOGONAL CONJUGATE OF GtX(31)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b6 - c6)/(b2 - c2)

X(3116) lies on these lines: 1,1581   31,48   38,75   63,1740   99,723   210,1575   982,2887   1932,2172

X(3116) = isogonal conjugate of X(3113)
X(3116) = crosspoint of X(75) and X(2186)
X(3116) = crossdifference of every pair of points on the line X(1577)X(1924)

### X(3117) = ISOGONAL CONJUGATE OF TgX(32)

Trilinears     a3(b4 + c4 + b2c2) : :

X(3117) lies on these lines: 2,39   3,695   6,694   32,184   99,707   110,737   187,353   982,2275   1334,3009

X(3117) = isogonal conjugate of X(3114)
X(3117) = crossdifference of every pair of points on line X(669)X(804)

### X(3118) = ISOGONAL CONJUGATE OF GtX(32)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(b8 - c8)/(b2 - c2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3118) lies on these lines: 32,184   76,141   83,694   99,695   2076,2916

X(3118) = isogonal conjugate of X(3115)

Danneels Perspectors

Suppose A1B1C1 is the cevian triangle of a point X.
Let LAB be the line through B parallel to A1B1, and let LAC be the line through C parallel to A1C1.
Let A2 = LAB∩LAC, and define B2 and C2 cyclically. Let

A3 = BB2∩CC2,    B3 = CC2∩AA2,    C3 = AA2∩BB2.

Eric Danneels proves in "A Simple Perspectivity," Forum Geometricorum 6 (2006) 199-203, that the triangles A3B3C3 and ABC are perspective. If X = x : y : z (barycentrics), then the Danneels perspector P(X) is given by

P(X) = x(y - z)2 : y(z - x)2 : z(x - y)2.

If X' denotes the isotomic conjugate of X, then P(X') = P(X). A few examples selected from Danneels's article follow:

P(X(1)) = P(X(75)) = X(244)
P(X(3)) = P(X(264)) = X(2972)
P(X(4)) = P(X(69)) = X(125)
P(X(7)) = P(X(8)) = X(11).

### X(3119) = DANNEELS PERSPECTOR FOR X(9)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)2(b + c - a)3
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3119) lies on these lines: 9,100   11,1146   125,661   281,2171   282,604   1200,1864

X(3119) = X(i)-Ceva conjugate of X(j) for these (i,j): (85,522), (282,663), (1146,2310)
X(3119) = X(3022)-cross conjugate of X(2310)
X(3119) = crosspoint of X(i) and X(j) for these (i,j): (33,650), (85,522), (514,3062)
X(3119) = crosssum of X(i) and X(j) for these (i,j): (41,109), (57,934), (77,651), (101,165), (269,1461)
X(3119) = crossdifference of every pair of points on the line X(109)X(934)

### X(3120) = DANNEELS PERSPECTOR FOR X(10)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c)(b - c)2
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3120) lies on these lines: 1,149   2,846   7,2648   11,244   27,2206   31,1836   38,2886   42,226   58,79   109,2006   115,125   225,1042   321,2887   334,1978   442,2292   516,902   614,990   726,3006   851,1284   899,908   946,1201   986,2476   1109,2632   1365,2611   1834,2650   2170,2969

X(3120) = reflection of X(902) in X(3011)
X(3120) = X(i)-Ceva conjugate of X(j) for these (i,j): (10,523), (27,649), (79,513), (86,514), (226,661), (313,1577), (1086,3125)
X(3120) = cevapoint of X(115) and X(2643)
X(3120) = X(2643)-cross conjugate of X(3125)
X(3120) = crosspoint of X(i) and X(j) for these (i,j): (10,523), (65,513), (86,514), (313,1577), (321,693), (1086,1111)
X(3120) = crosssum of X(i) and X(j) for these (i,j): (21,100), (42,101), (58,110), (163,2206), (692,1333), (1110,1252)
X(3120) = crossdifference of every pair of points on the line X(101)X(110)
X(3120) = isogonal conjugate of X(4570)
X(3120) = isotomic conjugate of X(4600)
X(3120) = intersection of tangents to Steiner inellipse at X(115) and X(1086)
X(3120) = crosspoint wrt medial triangle of X(115) and X(1086)
X(3120) = complement of X(4427)
X(3120) = X(10)-isoconjugate of X(1101)
X(3120) = barycentric product X(i)*X(j) for these {i,j}: {10,1086}, {514,423}

### X(3121) = DANNEELS PERSPECTOR FOR X(37)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b + c)(b - c)2
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3121) lies on these lines: 81,893   100,292   244,665   351,865   890,1977   1402,2205

X(3121) = isogonal conjugate of X(4601)
X(3121) = X(i)-Ceva conjugate of X(j) for these (i,j): (37,512), (213,798), (274,513), (893,649), (1015,3122), (1333,667), (1402,669), (2203,1980)
X(3121) = crosspoint of X(i) and X(j) for these (i,j): (37,512), (42,649), (213,798), (274,513), (667,1333)
X(3121) = crosssum of X(i) and X(j) for these (i,j): (81,99), (86,190), (100,213), (274,799), (314,645), (321,668)
X(3121) = crossdifference of every pair of points on the line X(99)X(100)
X(3121) = intersection of tangents to Steiner inellipse at X(1015) and X(1084)
X(3121) = crosspoint wrt medial triangle of X(1015) and X(1084)

### X(3122) = DANNEELS PERSPECTOR FOR X(42)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(b - c)2
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3122) lies on these lines: 6,2054   11,244   86,256   190,291   351,865   674,3009   1015,1960   1400,1918   1929,2640   2092,2667   2110,2277   2642,2643

X(3122) = X(i)-Ceva conjugate of X(j) for these (i,j): (10,661), (42,512), (58,649), (244,3125), (256,513), (310,514), (1015,3121), (1400,798), (1474,1919)
X(3122) = X(3124)-cross conjugate of X(3125)
X(3122) = crosspoint of X(i) and X(j) for these (i,j): (10,661), (37,513), (42,512), (58,649), (244,1015), (310,514), (667,1402)
X(3122) = crosssum of X(i) and X(j) for these (i,j): (10,190), (58,662), (81,100), (86,99), (101,1918), (306,1331), (314,668), (333,643), (645,1043), (765,1016), (1332,1792)
X(3122) = isogonal conjugate of X(4600)
X(3122) = intersection of tangents to Steiner inellipse at X(1084) and X(1086)
X(3122) = crosspoint wrt medial triangle of X(1084) and X(1086)
X(3122) = crossdifference of every pair of points on the line X(99)X(101)

### X(3123) = DANNEELS PERSPECTOR FOR X(43)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (bc - ab - ac)(b - c)2
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3123) lies on these lines: 11,244   31,1633   38,1227   350,2227   536,2228   661,1084   1423,2209

X(3123) = X(i)-Ceva conjugate of X(i) and X(j) for these (i,j): (76,661), (1015,244)
X(3123) = crosspoint of X(75) and X(513)
X(3123) = crosssum of X(i) and X(j) for these (i,j): (31,100), (87,932), (101,2209)
X(3123) = crossdifference of every pair of points on the line X(101,932)

### X(3124) = DANNEELS PERSPECTOR FOR X(76)

Trilinears    a(b2 - c2)2 : :

X(3124) lies on the Brocard inellipse and these lines: 2,694   6,110   23,1691   25,1501   32,3457   39,373   42,2054   51,1196   115,125   237,2021   244,661 nbsp;  351,865   394,2987   593,2248   1193,2653   1495,1692   1506,3118   1613,3060   1994,2056   2092,3030   2501,2970   2679,3005

X(3124) = isogonal conjugate of X(4590)
X(3124) = complement of X(4576)
X(3124) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,3005), (6,512), (26,669), (76,523), (83,2514), (111,351), (755,2872), (2207,2489), (2248,649), (3125,2643)
X(3124) = X(i)-cross conjugate of X(j) for these (i,j): (1084,2971), (2679,2086)
X(3124) = crosspoint of X(i) and X(j) for these (i,j): (6,512), (25,2501), (76,523), (647,2351), (661,756), (2207,2489), (3122,3125)
X(3124) = crosssum of X(i) and X(j) for these (i,j): (2,99), (6,1634), (32,110), (112,459), (317,648), (662,757), (6189,6190)
X(3124) = X(i)-line conjugate of X(j) for these (i,j): 6,110   111,110
X(3124) = crossdifference of every pair of points on the line X(99)X(110)
X(3124) = perspector of circumconic centered at X(3005)
X(3124) = center of circumconic that is locus of trilinear poles of lines passing through X(3005)
X(3124) = trilinear pole wrt symmedial triangle of Brocard axis
X(3124) = intersection of tangents to Steiner inellipse at X(115) and X(1084)
X(3124) = crosspoint wrt medial triangle of X(115) and X(1084)
X(3124) = bicentric difference of PU(105)
X(3124) = PU(105)-harmonic conjugate of X(351)
X(3124) = perspector of unary cofactor triangles of 1st and 2nd Brocard triangles

### X(3125) = DANNEELS PERSPECTOR FOR X(81)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b - c)2
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3125) lies on these lines: 1,1929   6,1718   10,762   37,1018   38,1573   65,213   115,125   244,665   257,274   335,668   614,1572   758,2238   910,2251   918,1086   1333,2160   1411,1415   1575,1739   2092,2294   2642,2643

X(3125) = isogonal conjugate of X(4567)
X(3125) = isotomic conjugate of X(4601)
X(3125) = X(i)-Ceva conjugate of X(j) for these (i,j): (28,667), (37,661), (65,512), (81,513), (244,3122), (257,514), (274,2530), (321,523), (1086,3120), (2160,649)
X(3125) = cevapoint of X(2643) and X(3124)
X(3125) = X(i)-cross conjugate of X(j) for these (i,j): (2643,3120), (3124,3122)
X(3125) = crosspoint of X(i) and X(j) for these (i,j): (10,514), (37,661), (81,513), (244,1086), (321,523), (649,1400), (1880,2501)
X(3125) = crosssum of X(i) and X(j) for these (i,j): (37,100), (58,101), (72,906), (81,662), (110,1333), (190,333), (643,2287), (692,2205), (765,1252), (1331,2327)
X(3125) = intersection of tangents to Steiner inellipse at X(115) and X(1015)
X(3125) = crosspoint wrt medial triangle of X(115) and X(1015)
X(3125) = trilinear product of extraversions of X(37)
X(3125) = crossdifference of every pair of points on the line X(100)X(110)

### X(3126) = DANNEELS PERSPECTOR FOR X(100)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)(b2 + c2 - ab - ac)2
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3126) lies on these lines: 1,905   2,885   3,667   9,513   10,514   119,120   142,522   442,1577   650,1027   656,2092   665,1642   1025,1026

### X(3127) = X(4)-CEVA CONJUGATE OF X(1162)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(4) and u : v : w = X(1162)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b2 + c2 + S)/(b2 + c2 - a2)     (E. Danneels)

As a point on the Euler line, X(3127) has Shinagawa coefficients (F,E + F + S).

X(3127) lies on these lines: 2,3   1162,1165   1587,1899   1853,3070

X(3127) = X(4)-Ceva conjugate of X(1162)

### X(3128) = X(4)-CEVA CONJUGATE OF X(1163)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(4) and u : v : w = X(1163)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b2 + c2 - S)/(b2 + c2 - a2)     (E. Danneels)

As a point on the Euler line, X(3128) has Shinagawa coefficients (F,E + F - S).

X(3128) lies on these lines: 2,3   1163,1164   1588,1899   1853,3071

X(3128) = X(4)-Ceva conjugate of X(1163)

### X(3129) = X(13)-CEVA CONJUGATE OF X(6)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(13) and u : v : w = X(6)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

As a point on the Euler line, X(3129) has Shinagawa coefficients (31/2F - S, -31/2E - 31/2F + S).

X(3129) lies on these lines: 2,3   13,1605   15,1495   51,62   61,184   619,2926   1251,1953

X(3129) = isogonal conjugate of X(2992)
X(3129) = X(13)-Ceva conjugate of X(6)
X(3129) = pole wrt circumcircle of trilinear polar of X(13) (line X(395)X(523))
X(3129) = crosspoint of circumcircle intercepts of inner Napoleon circle

### X(3130) = X(14)-CEVA CONJUGATE OF X(6)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(14) and u : v : w = X(6)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

As a point on the Euler line, X(3130) has Shinagawa coefficients (31/2F + S, -31/2E - 31/2F - S).

X(3130) lies on these lines: 2,3   14,1606   16,1495   51,61   62,184   618,2925

X(3130) = isogonal conjugate of X(2993)
X(3130) = X(14)-Ceva conjugate of X(6)
X(3130) = pole wrt circumcircle of trilinear polar of X(14) (line X(396)X(523))
X(3130) = crosspoint of circumcircle intercepts of outer Napoleon circle

### X(3131) = X(17)-CEVA CONJUGATE OF X(6)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(17) and u : v : w = X(6)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

As a point on the Euler line, X(3131) has Shinagawa coefficients (F - 31/2S, -E - F + 31/2S).

X(3131) lies on these lines: 2,3   15,184   16,51   17,1607

X(3131) = X(17)-Ceva conjugate of X(6)

### X(3132) = X(18)-CEVA CONJUGATE OF X(6)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(18) and u : v : w = X(6)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

As a point on the Euler line, X(3132) has Shinagawa coefficients (F + 31/2S, -E - F - 31/2S).

X(3132) lies on these lines: 2,3   15,51   16,184   18,1608

X(3132) = X(18)-Ceva conjugate of X(6)

### X(3133) = X(24)-CEVA CONJUGATE OF X(52)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(24) and u : v : w = X(52)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

As a point on the Euler line, X(3133) has Shinagawa coefficients ((E + 2F)(E - 2F) - 4S2,(E + 2F)2 + 4S2).

X(3133) lies on these lines: 2,3      571,1147

X(3133) = X(24)-Ceva conjugate of X(52)

### X(3134) = X(74)-CEVA CONJUGATE OF X(523)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(74) and u : v : w = X(523)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

As a point on the Euler line, X(3134) has Shinagawa coefficients ((7E - 2F)F - 2S2,(E + F)(E - 2F) - 2S2).

X(3134) lies on these lines: 2,3   122.136   125,526   2970,2972

X(3134) = inverse-in-nine-point circle of X(3154)
X(3134) = X(74)-Ceva conjugate of X(523)
X(3134) = crosspoint of X(264) and X(2394)
X(3134) = crosssum of X(184) and X(2420)

### X(3135) = X(96)-CEVA CONJUGATE OF X(6)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(96) and u : v : w = X(6)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

As a point on the Euler line, X(3135) has Shinagawa coefficients ((E + 2F)F + 2S2, -(E + F)(E + 2F) - 2S2).

X(3135) lies on these lines: 2,3   51,570   160,184   161,2351

X(3135) = X(96)-Ceva conjugate of X(6)
X(3135) = crosssum of X(338) and X(924)

### X(3136) = X(101)-CEVA CONJUGATE OF X(523)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(101) and u : v : w = X(523)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)
X(3136) = 3a2b2c2*X(2) + 4S2(bc + ca + ab)*X(5)

As a point on the Euler line, X(3136) has Shinagawa coefficients (\$aSASB\$-\$aSCSA\$ +\$aSC\$F-\$aSB\$F, \$aSC2\$ -\$aSB2\$).

X(3136) lies on these lines: 2,3   12,42   71,1213   118,125   310,325   672,1901   1211,2886   1214,1893   1230,3006   1836,2245   2238,2911

X(3136) = inverse-in-orthocentroidal circle of X(1011)
X(3136) = X(102)-Ceva conjugate of X(523)
X(3136) = complement of X(4148)
X(3136) = crosspoint of X(10) and X(264)
X(3136) = crosssum of X(58) and X(184)

### X(3137) = X(102)-CEVA CONJUGATE OF X(523)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(102) and u : v : w = X(523)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

As a point on the Euler line, X(3137) has Shinagawa coefficients (\$aSA6\$ -\$aSA3\$[(E+F)3-3ES2] +\$aSBSC\$(E+F)FS2-\$a\$F2S4, -\$aSA4\$S2 +\$aSA3\$(E+F)S2+\$aSA\$FS4 -\$a\$(E+F)FS4).

X(3137) lies on these lines: 2,3   124,125

X(3137) = X(102)-Ceva conjugate of X(523)

### X(3138) = X(103)-CEVA CONJUGATE OF X(523)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(103) and u : v : w = X(523)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

As a point on the Euler line, X(3138) has Shinagawa coefficients (\$aSA5\$ -\$aSA2\$[(E+F)3-3ES2] +\$aSBSC\$ES2 +\$aSA\$(E+F)FS2-\$a\$(E+F)2FS2, -\$aSA3\$S2 +\$aSA2\$(E+F)S2 -\$aSBSC\$(E+F)S2+\$a\$FS4).

X(3138) lies on these lines: 2,3   116,125   2972,2973

X(3138) = X(103)-Ceva conjugate of X(523)

### X(3139) = X(104)-CEVA CONJUGATE OF X(523)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(104) and u : v : w = X(523)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

As a point on the Euler line, X(3139) has Shinagawa coefficients (2\$aSBSC\$+3\$aSA\$F-\$a\$S2, \$aSA\$(E+F)-\$a\$S2).

X(3139) lies on these lines: 2,3   11,125   123,136   2968,2970   2969,2972

X(3139) = X(104)-Ceva conjugate of X(523)

### X(3140) = X(105)-CEVA CONJUGATE OF X(523)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(105) and u : v : w = X(523)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

As a point on the Euler line, X(3140) has Shinagawa coefficients (\$abSC3\$+\$abSASB\$(E+F) -\$abSC\$[(E+F)2-2S2]-\$ab\$FS2, 3\$abSC\$S2-\$ab\$(E+F)S2).

X(3140) lies on these lines: 2,3   11,115   125,2775   339,2969   2968,2971

X(3140) = X(105)-Ceva conjugate of X(523)

### X(3141) = X(106)-CEVA CONJUGATE OF X(523)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(106) and u : v : w = X(523)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

As a point on the Euler line, X(3141) has Shinagawa coefficients (\$aSA4\$ -\$aSA2\$[(E+F)2-2S2] +2\$aSBSC\$S2 +2\$aSA\$FS2-2\$a\$(E+F)FS2, 3\$aSA2\$S2 -\$a\$[(E+F)2-2S2]S2).

X(3141) lies on these lines: 2,3   125,2776

X(3141) = X(106)-Ceva conjugate of X(523)

### X(3142) = X(109)-CEVA CONJUGATE OF X(523)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(109) and u : v : w = X(523)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

As a point on the Euler line, X(3142) has Shinagawa coefficients (\$aSA2SB\$ -\$aSCSA2\$-\$aSASB\$F +\$aSCSA\$F, \$aSB2SC\$ -\$aSBSC2\$-\$aSB\$S2 +\$aSC\$S2).

X(3142) lies on these lines: 2,3   11,1193   12,73   117,125   226,1425   1211,1329   1213,2183   1214,1867   1400,1901

X(3142) = X(109)-Ceva conjugate of X(523)
X(3142) = crosspoint of X(226) and X(264)
X(3142) = crosssum of X(184) and X(284)

### X(3143) = X(111)-CEVA CONJUGATE OF X(523)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(111) and u : v : w = X(523)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

As a point on the Euler line, X(3143) has Shinagawa coefficients (2(E + F)2F - (E - 2F)S2,2(E + F)3 - (7E - 2F)S2).

X(3143) lies on these lines: 2,3   115,804   125,2780   339,2971

X(3143) = X(111)-Ceva conjugate of X(523)
X(3143) = crossdifference of every pair of points on the line X(647)X(1634)

### X(3144) = X(225)-CEVA CONJUGATE OF X(4)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(225) and u : v : w = X(4)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = [a3 + 2a2(b + c) + abc - b3 - c3]/(b2 + c2 - a2)      (E. Danneels)

As a point on the Euler line, X(3144) has Shinagawa coefficients (2FS2, -ES2 - \$bcSBSC\$).

X(3144) lies on these lines: 2,3   34,978   158,2752   225,1247   242,1838   1068,1148   1193,1870   1714,3186

X(3144) = X(i)-Ceva conjugate of X(j) for these (i,j): (225,4), (1940,1148), (2907,1046)
X(3144) = X(2907)-cross conjugate of X(4)

### X(3145) = X(226)-CEVA CONJUGATE OF X(6)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(226) and u : v : w = X(6)
Trilinears        g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a[(a5 - a3(b2 + bc + c2) + a2(b3 + c3) + abc(b + c)2 - (b + c)(b2 + bc + c2)(b - c)2]      (E. Danneels)
Barycentrics   ag(a,b,c) : bg(b,c,a) : cg(c,a,b)

X(3145) lies on these lines: 1,1283   2,3   35,228   41,2276   45,198   51,580   55,976   56,1626   109,1425   184,581   283,511   500,1437   692,2594   1284,1486   1400,1950   1495,2360

X(3145) = X(226)-Ceva conjugate of X(6)
X(3145) = crosspoint of X(109) and X(250)
X(3145) = crosssum of X(125) and X(522)

### X(3146) = X(253)-CEVA CONJUGATE OF X(2)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(253) and u : v : w = X(2)
= 4 cos A - 3 sin B sin C : 4 cos B - 3 sin C sin A : 4 cos C - 3 sin A sin B
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 5a4 - 2a2(b2 + c2) - 3(b2 - c2)2      (E. Danneels)

As a point on the Euler line, X(3146) has Shinagawa coefficients (1, -4).

The point X(3146) exemplifies a theorem based on a discussion in Tran Quang Hung and Peter Moses, Hyacinthos 25668. The theorem (Peter Moses, April 10, 2017) is as follows.

Suppose that P is a point on a circumhyperbola H. Let A'B'C' be the cevian triangle of P and let O' be the circumcircle of A'B'C'. Let A'' be the point other than A' that lies on both O' and line BC, and define B'' and C'' cyclically. The orthocenter of A"B"C" lies on the isogonal conjugate of H.

Writing P' for the orthocenter of of A"B"C", examples of the mapping P -> P' include the following:
Jerabek hyperbola -> Euler line
Feuerbach hyperbola -> X(1)X(3)
Kiepert hyperbola -> Brocard axis.

The image of a point P = a^2 / (S^2+k SB SC) : : on the Jerabek hyperbola is the following point on the Euler line:

(S^2 SA SB+S^2 SB^2+S^2 SA SC+4 S^2 SB SC+3 k SA SB^2 SC+S^2 SC^2+3 k SA SB SC^2+2 k SB^2 SC^2) (S^4 SA^3+S^4 SA^2 SB+S^4 SA SB^2-2 k S^2 SA^3 SB^2-k^2 SA^4 SB^3+S^4 SA^2 SC+7 S^4 SA SB SC-2 k S^2 SA^3 SB SC+2 S^4 SB^2 SC+8 k S^2 SA^2 SB^2 SC-3 k^2 SA^4 SB^2 SC+2 k S^2 SA SB^3 SC+k^2 SA^3 SB^3 SC+S^4 SA SC^2-2 k S^2 SA^3 SC^2+2 S^4 SB SC^2+8 k S^2 SA^2 SB SC^2-3 k^2 SA^4 SB SC^2+14 k S^2 SA SB^2 SC^2+3 k^2 SA^3 SB^2 SC^2+2 k S^2 SB^3 SC^2+7 k^2 SA^2 SB^3 SC^2-k^2 SA^4 SC^3+2 k S^2 SA SB SC^3+k^2 SA^3 SB SC^3+2 k S^2 SB^2 SC^3+7 k^2 SA^2 SB^2 SC^3+5 k^2 SA SB^3 SC^3)

For P = X(69), the parameter k = = -SW S^2 / (SA SB SC), and P' = X(3146)
For P = X(68), see X(13322).

X(3146) lies on the cubics K117, K127, K347, K425, K461, K558, K841, and these lines: {1,5556}, {2,3}, {7,950}, {8,144}, {10,9778}, {11,5265}, {12,5281}, {13,5366}, {14,5365}, {33,4296}, {34,3100}, {35,10590}, {36,10591}, {40,3219}, {51,10574}, {52,6241}, {55,5229}, {56,5225}, {61,5335}, {62,5334}, {63,5175}, {65,9961}, {69,11469}, {74,12295}, {78,1750}, {84,3218}, {107,5896}, {145,515}, {146,6193}, {147,7900}, {148,2794}, {149,2829}, {150,10725}, {151,10726}, {152,10727}, {153,5840}, {154,5893}, {165,9780}, {185,3060}, {193,1503}, {226,4313}, {265,12244}, {279,4872}, {280,5081}, {315,10513}, {316,3926}, {317,6527}, {323,11441}, {329,3984}, {346,7270}, {347,7282}, {355,4678}, {385,2996}, {388,390}, {389,11002}, {393,3284}, {485,6453}, {486,6454}, {489,1271}, {490,1270}, {497,3304}, {498,4324}, {499,4316}, {511,5921}, {517,3621}, {519,9589}, {527,12625}, {528,12632}, {542,8596}, {543,7758}, {576,6776}, {578,11003}, {671,10991}, {938,3586}, {942,11220}, {944,3623}, {946,3622}, {971,3868}, {990,5262}, {1056,9655}, {1058,9668}, {1105,1629}, {1125,9779}, {1131,3068}, {1132,3069}, {1151,2671}, {1152,2672}, {1173,4846}, {1181,1994}, {1219,4514}, {1285,5305}, {1327,8960}, {1350,3620}, {1351,12174}, {1352,7929}, {1376,8165}, {1467,8544}, {1478,3746}, {1479,4293}, {1493,12254}, {1495,11449}, {1498,1993}, {1539,12121}, {1587,6419}, {1588,6420}, {1614,9545}, {1697,8545}, {1698,12512}, {1699,3616}, {1737,4333}, {1836,3486}, {1837,3474}, {1853,5894}, {1870,9538}, {1891,4329}, {1968,10313}, {2548,7756}, {2549,7747}, {2777,3448}, {2883,11206}, {2979,5907}, {3047,9934}, {3055,11742}, {3058,9657}, {3070,3592}, {3071,3594}, {3085,3585}, {3086,3583}, {3087,5158}, {3101,11471}, {3180,5869}, {3181,5868}, {3241,4301}, {3306,9841}, {3329,5395}, {3346,10152}, {3431,5944}, {3436,6253}, {3476,12701}, {3488,11036}, {3579,5818}, {3580,5925}, {3590,10147}, {3591,10148}, {3601,5226}, {3618,10541}, {3624,12571}, {3672,5716}, {3679,5493}, {3681,7957}, {3785,7802}, {3812,5918}, {3817,5550}, {3818,10519}, {3869,12688}, {3870,12651}, {3873,12680}, {3876,5927}, {3877,9856}, {3935,6769}, {4295,10572}, {4298,10580}, {4304,5703}, {4305,12047}, {4308,12053}, {4309,5270}, {4311,9614}, {4312,6738}, {4314,5290}, {4317,4857}, {4325,10072}, {4330,10056}, {4355,6744}, {4440,11851}, {4855,5748}, {4907,7273}, {5007,5286}, {5012,11424}, {5032,8550}, {5057,12679}, {5080,5537}, {5128,7285}, {5204,10589}, {5217,10588}, {5218,10895}, {5250,11372}, {5251,12511}, {5254,5304}, {5260,5584}, {5284,8273}, {5302,7964}, {5319,11648}, {5328,5438}, {5434,9670}, {5435,9581}, {5446,5890}, {5480,7864}, {5596,9968}, {5609,7728}, {5640,9729}, {5643,9815}, {5656,9716}, {5658,11015}, {5663,6243}, {5698,5794}, {5734,5882}, {5763,9963}, {5787,12246}, {5795,11530}, {5806,10167}, {5842,12667}, {5870,6280}, {5871,6279}, {5889,6000}, {6054,10992}, {6194,6248}, {6256,10528}, {6321,9862}, {6337,7773}, {6427,7581}, {6428,7582}, {6482,12818}, {6483,12819}, {6515,12324}, {6519,8981}, {6564,9540}, {6759,9544}, {6800,11425}, {7288,10896}, {7710,7783}, {7712,11430}, {7738,7745}, {7785,8721}, {7787,12203}, {7795,7842}, {7928,10516}, {7989,10164}, {8117,8122}, {8118,8121}, {9140,10990}, {9580,9785}, {9613,10624}, {9730,9781}, {9748,12110}, {9799,9965}, {10404,11038}, {10446,10454}, {10453,12545}, {10625,11459}, {10711,10993}, {10738,12248}, {10749,12253}, {11004,11456}, {11411,12293}, {11412,11455}, {11416,11470}, {11417,11473}, {11418,11474}, {11420,11475}, {11421,11476}, {11432,11820}, {11523,12536}, {11750,12897}, {12160,12315}, {12219,12292}, {12220,12294}, {12223,12298}, {12224,12299}, {12226,12300}, {12282,12290}, {12317,12902}, {12684,12690}

X(3146) = midpoint of X(i) in X(j) for these {i,j}: {382,5073}, {3529,11541}
X(3146) = reflection of X(i) in X(j) for these (i,j): (2,3543), (3,3627), (4,382), (8,5691), (20,4), (74,12295), (145,962), (146,10721), (147,10722), (148,10723), (149,10724), (150,10725), (151,10726), (152,10727), (153,10728), (376,3830), (550,3853), (944,12699), (1657,5), (3448,10733), (3529,3), (3869,12688), (5059,20), (5189,10296), (5925,6247), (5984,148), (6225,5895), (6240,7553), (6241,52), (6361,355), (9862,6321), (9961,65), (10430,3586), (10575,5446), (11001,381), (11411,12293), (11412,12162), (11750,12897), (12103,12102), (12111,11381), (12121,1539), (12219,12292), (12220,12294), (12221,12296), (12222,12297), (12223,12298), (12224,12299), (12225,1885), (12226,12300), (12244,265), (12246,5787), (12248,10738), (12253,10749), (12279,185), (12317,12902), (12383,7728), (12536,11523)
X(3146) = isogonal conjugate of X(3532)
X(3146) = complement of X(5059)
X(3146) = anticomplement of X(20)
X(3146) = X(253)-Ceva conjugate of X(2)
X(3146) = crosspoint of X(1131) and X(1132)
X(3146) = crosspoint of PU(i) for these i: 116, 117
X(3146) = crosssum of X(1151) and X(1152)
X(3146) = orthocentroidal-circle-inverse of X(3832)
X(3146) = intersection of tangents to Kiepert hyperbola at X(1131) and X(1132)
X(3146) = orthocenter of cevian triangle of X(253)
X(3146) = X(11531)-of-orthic-triangle if ABC is acute
X(3146) = exsimilicenter of polar circle and de Longchamps circle
X(3146) = Ehrmann-mid-to-Johnson similarity image of X(382)
X(3146) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,4,3832), (2,20,3522), (2,3832,5068), (2,3854,5), (2,5059,20), (2,6872,11106), (3,4,3091), (3,5,3525), (3,381,3628), (3,382,3627), (3,546,3090), (3,1657,12103), (3,3090,10303), (3,3091,2), (3,3525,3523), (3,3529,20), (3,3627,4), (3,3628,631), (3,3843,5079), (3,5072,632), (3,5076,546), (3,5079,140), (3,5198,1995), (3,7530,3518), (3,12103,376), (4,5,3839), (4,20,2), (4,21,6870), (4,24,6623), (4,376,5), (4,377,6894), (4,382,3543), (4,411,6871), (4,550,5056), (4,631,381), (4,1593,7378), (4,1657,3523), (4,3090,546), (4,3522,5068), (4,3523,3854), (4,3524,3855), (4,3528,3545), (4,3529,3), (4,3533,3858), (4,3534,7486), (4,3545,3843), (4,3575,6995), (4,3651,6843), (4,3855,3845), (4,5059,3522), (4,6240,7487), (4,6622,10151), (4,6815,7394), (4,6836,5046), (4,6850,6839), (4,6851,6840), (4,6865,6957), (4,6868,6837), (4,6869,6838), (4,6875,6866), (4,6876,6867), (4,6897,6849), (4,6899,6893), (4,6903,6929), (4,6906,6844), (4,6909,5187), (4,6916,6835), (4,6925,2475), (4,6934,6848), (4,6938,6847), (4,6942,6968), (4,6948,6953), (4,6988,7548), (4,7580,5177), (4,7714,1906), (4,10299,3850), (4,10431,6895), (4,10996,6997), (4,11001,631), (4,11111,10883), (4,11541,3529), (5,376,3523), (5,550,12100), (5,1657,376), (5,3523,2), (5,3627,12102), (5,3830,4), (5,3839,3854), (5,10124,1656), (5,12103,3), (20,1559,2060), (20,3091,3), (20,3523,376), (20,3543,4), (20,3830,3854), (20,3839,3523), (20,4208,7411), (20,5056,10304), (20,7396,11413), (20,7486,3528), (20,10304,550), (21,5177,2), (22,7378,2), (23,12086,3), (24,12085,2071), (25,7396,2), (26,3520,10298), (55,5229,5261), (56,5225,5274), (140,3534,3528), (140,3545,7486), (140,3843,3545), (140,3857,5079), (140,3860,5), (140,7486,2), (376,1657,20), (376,3525,3), (376,3830,3839), (376,3839,2), (376,12100,10304), (376,12102,3091), (377,452,2), (377,11114,452), (378,7387,7488), (381,550,631), (381,631,5056), (381,3853,4), (381,10304,2), (381,11001,10304), (382,1657,3830), (382,11541,3091), (388,6284,390), (404,6919,2), (405,4208,2), (411,1012,4189), (427,10565,2), (428,7386,7398), (443,5129,2), (443,11113,5129), (489,12323,1271), (490,12322,1270), (497,7354,3600), (546,632,5072), (546,3090,3091), (546,3627,5076), (546,5076,4), (546,12103,12108), (546,12108,5), (548,1656,3524), (548,3845,1656), (549,3851,5067), (549,3861,3851), (550,631,10304), (550,3628,3), (550,3853,381), (550,11001,20), (631,5056,2), (631,11001,550), (631,12100,3523), (632,5072,3090), (858,4232,2), (946,5731,3622), (950,9579,7), (962,6223,5905), (1006,6993,2), (1327,9681,8960), (1370,6995,2), (1479,10483,4293), (1532,6943,5154), (1597,11414,7503), (1656,3845,3855), (1657,3830,5), (1657,3854,3522), (1657,3860,3528), (1657,12102,3525), (1699,4297,3616), (1885,12173,4), (1907,6823,5133), (2043,2044,3524), (2071,6623,2), (2475,6872,2), (2478,6904,2), (2979,11439,5907), (3060,12279,185), (3070,6459,7585), (3071,6460,7586), (3090,10303,2), (3091,10303,3090), (3091,11541,5059), (3149,6909,4188), (3151,6994,2), (3152,7518,2), (3515,10151,6622), (3518,7464,3), (3522,3832,2), (3523,3839,5), (3523,10303,12108), (3524,3855,1656), (3525,3529,12103), (3525,12102,3839), (3525,12108,10303), (3526,3850,5071), (3526,8703,10299), (3528,3545,140), (3528,3843,7486), (3529,3627,3091), (3529,5076,10303), (3529,12102,3523), (3530,3858,5055), (3530,5055,3533), (3530,12101,3858), (3534,3830,3860), (3534,3843,140), (3534,5079,3), (3543,3839,3830), (3543,5056,3853), (3543,5059,3832), (3543,10303,5076), (3545,3857,3091), (3583,4299,3086), (3585,4302,3085), (3586,4292,938), (3616,10248,1699), (3627,5073,11541), (3627,11541,20), (3627,12102,3830), (3817,7987,5550), (3839,3854,3832), (3843,5079,3857), (3850,8703,3526), (3853,11001,5056), (3855,12811,3091), (3857,5079,3545), (4188,5187,2), (4189,6871,2), (4190,5046,2), (4304,9612,5703), (4314,5290,10578), (5004,5005,9909), (5056,10304,631), (5071,10299,3526), (5189,7519,2), (5446,10575,5890), (6039,6040,10011), (6284,12943,388), (6825,6888,2), (6826,6992,2), (6831,6932,5141), (6833,6960,2), (6834,6972,2), (6837,6908,2), (6838,6847,2), (6839,6987,2), (6848,6890,2), (6850,7491,6987), (6868,6923,6908), (6869,6938,20), (6884,6889,2), (6885,6929,6964), (6891,6979,2), (6917,6930,6846), (6926,6953,2), (6928,6948,6926), (6999,7406,2), (7000,7374,1513), (7354,12953,497), (7379,7390,2), (7386,7398,2), (7391,7500,2), (7503,11414,6636), (7517,12084,186), (7526,12083,7512), (7527,12082,7492), (7737,7748,5286), (7802,11185,3785), (9580,10106,9785), (11412,11455,12162), (12102,12103,5), (12102,12108,546)

### X(3147) = X(254)-CEVA CONJUGATE OF X(4)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(254) and u : v : w = X(4)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

As a point on the Euler line, X(3147) has Shinagawa coefficients (4F, -E - 2F).

X(3147) lies on these lines: 2,3   323,2904   1068,1940   1249,3003

X(3147) = X(254)-Ceva conjugate of X(4)

### X(3148) = X(262)-CEVA CONJUGATE OF X(6)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(262) and u : v : w = X(6)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

As a point on the Euler line, X(3148) has Shinagawa coefficients ((E + F )F - S2, -(E + F)2 + S2).

X(3148) lies on these lines: 2,3   6,157   32,51   39,184   206,570   216,1974   574,1495   577,1843   878,1640   1993,2001

X(3148) = inverse-in-orthocentroidal circle of X(2450)
X(3148) = X(262)-Ceva conjugate of X(6)
X(3148) = crossdifference of every pair of points on the line X(647)X(2799)

},

### X(3149) = X(285)-CEVA CONJUGATE OF X(1)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(285) and u : v : w = X(1)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

As a point on the Euler line, X(3149) has Shinagawa coefficients (\$aSA\$ - abc, -\$aSA\$ - abc).

X(3149) lies on these lines: 1,227   2,3   35,1699   40,936   55,946   56,515   57,1071   64,1715   78,517   84,1728   100,962   221,1771   222,1745   282,2270   355,956   573,1437   578,1437   581,940   603,2635   908,1259   938,944   971,1445   1155,1158   1454,1858   1498,1754   1617,3086   1735,1854   1741,1903

X(3149) = X(285)-Ceva conjugate of X(1)
X(3149) = X(21)-gimel conjugate of X(3)

### X(3150) = X(290)-CEVA CONJUGATE OF X(525)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(290) and u : v : w = X(525)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

As a point on the Euler line, X(3150) has Shinagawa coefficients (2(E + F)(2E - F)F - (E + 2F)S2, -2(E + F)2F + (E - 2F)S2).

X(3150) lies on these lines: 2,3   115,122   125,127   339,2972   879,2435

X(3150) = X(i)-Ceva conjugate of X(j) for these (i,j): (290,525), (1297,523)
X(3150) = crossdifference of X(i) and X(j) for every pair of points on the line X(647)X(1624)

### X(3151) = X(306)-CEVA CONJUGATE OF X(2)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(306) and u : v : w = X(2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

As a point on the Euler line, X(3151) has Shinagawa coefficients (E + \$bc\$, -2E - 2F - 2\$bc\$).

X(3151) lies on these lines: 2,3   63,2893   71,1654   144,2895   152,2822   306,1330

X(3151) = reflection of X(27) in X(440)
X(3151) = anticomplement of X(27)
X(3151) = X(i)-Ceva conjugate of X(j) for these (i,j): (306,2), (1330,1654)

### X(3152) = X(307)-CEVA CONJUGATE OF X(2)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(307) and u : v : w = X(2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

As a point on the Euler line, X(3152) has Shinagawa coefficients ((E + F + \$bc\$)\$aSBSC\$ + (\$a\$F + \$(b+c)\$)S2, 2\$a\$(E + F + \$bc\$)S2).

X(3152) lies on these lines: 2,3   40,151   78,1330   145,347   307,2893   1654,3177   2654,3100

X(3152) = anticomplement of X(29)
X(3152) = X(307)-Ceva conjugate of X(2)

### X(3153) = X(328)-CEVA CONJUGATE OF X(2)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(328) and u : v : w = X(2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

As a point on the Euler line, X(3153) has Shinagawa coefficients (E, -2E - 8F).

X(3153) lies on these lines: 2,3   110,1568   146,1531   265,1154   827,2697   1287,1297

X(3153) = refection of X(i) in X(j) for these (i,j): (20,2071), (23,403), (110,1568), (186,2072), (2070,5), (2071,858)
X(3153) = anticomplement of X(186)
X(3153) = inverse-in-de-Longchamps-circle of X(3)
X(3153) = inverse-in-Johnson-circle of X(5)
X(3153) = homothetic center of dual of orthic triangle and Ehrmann vertex-triangle
X(3153) = X(484)-of-Ehrmann-vertex-triangle

### X(3154) = X(477)-CEVA CONJUGATE OF X(523)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(477) and u : v : w = X(523)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

As a point on the Euler line, X(3154) has Shinagawa coefficients (9(E - 2F)F - 2S2, -(E + F)(E + 10 F) + 6S2).

X(3154) lies on these lines: 2,3   125,523

X(3154) = midpoint of X(125) and X(3258)
X(3154) = inverse-in nine-point circle of X(3134)
X(3154) = X(477)-Ceva conjugate of X(523)
X(3154) = crossdifference of every pair of point on the line X(647)X(2420)
X(3154) = orthogonal projection of X(125) on Euler line
X(3154) = Euler line intercept of Simpson line of X(74)

### X(3155) = X(485)-CEVA CONJUGATE OF X(6)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(485) and u : v : w = X(6)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

As a point on the Euler line, X(3155) has Shinagawa coefficients (F - S, -E - F + S).

X(3155) lies on these lines: 2,3   51,372   154,1151   157,590   184,371   485,2351   493,1976

X(3155) = X(i)-Ceva conjugate of X(j) for these (i,j): (485,6), (2351,3156)

### X(3156) = X(486)-CEVA CONJUGATE OF X(6)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(486) and u : v : w = X(6)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

As a point on the Euler line, X(3156) has Shinagawa coefficients (F + S, -E - F - S).

X(3156) lies on these lines: 2,3   51,371   154,1152   157,615   184,372   486,2351   494,1976

X(3156) = X(i)-Ceva conjugate of X(j) for these (i,j): (486,6), (2351,3155)

### X(3157) = X(1)-CEVA CONJUGATE OF X(3)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(1) and u : v : w = X(3)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3157) lies on these lines: 1,90   3,73   6,169   12,68   31,1066   40,1419   46,1079   55,500   56,215   65,921   72,394   221,517   495,611   602,1458   916,2293   971,1498   999,1201   1062,1071   1064,1496   1092,1425   1339,1616   1433,1807   1480,3057   1745,1936   1854,2771   1870,1993

X(3157) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,3), (3173,3211), (3193,46)
X(3157) = crosspoint of X(i) and X(j) for these (i,j): (1,46), (1800,3193)
X(3157) = crosssum of X(1) and X(90)

### X(3158) = X(1)-CEVA CONJUGATE OF X(9)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b + c - 3a)     (M. Iliev, 5/13/07)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3158) lies on these lines: 1,474   9,55   31,678   40,758   41,728   42,1449   57,100   78,1697   145,1420   165,518   528,1699   612,1962   1743,3052   1998,2078   2340,3208

X(3158) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,9), (145,1743)
X(3158) = crosspoint of X(i) and X(j) for these (i,j): (1,1743), (145,3161)
X(3158) = crosssum of X(513) and X(3020)

### X(3159) = X(1)-CEVA CONJUGATE OF X(10)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(1) and u : v : w = X(10)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3159) lies on these lines: 10,321   37,39   58,190   72,519   192,386

X(3159) = midpoint of X(72) and X(2901)
X(3159) = reflection of X(596) in X(1125)
X(3159) = X(1)-Ceva conjugate of X(10)
X(3159) = crosspoint of X(1) and X(3216)

### X(3160) = X(2)-CEVA CONJUGATE OF X(7)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(2) and u : v : w = X(7)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3160) = perspector of triangle ABC and the tangential triangle of the conic that passes through A, B, C and has center X(7)

X(3160) lies on these lines: 1,7   3,934   8,348   9,2124   144,1419   220,651   241,2275   738,1697   944,1565   1358,1388

X(3160) = midpoint of X(175) and X(176)
X(3160) = X(2)-Ceva conjugate of X(7)
X(3160) = cevapoint of X(i) and X(j) for these (i,j): (1,2124), (165,1419)
X(3160) = X(165)-cross conjugate of X(144)
X(3160) = crosspoint of X(2) and X(144)
X(3160) = crosssum of X(663) and X(3022)
X(3160) = X(99)-beth conjugate of X(8)

### X(3161) = X(2)-CEVA CONJUGATE OF X(8)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c - a)(b + c - 3a)     (M. Iliev, 5/13/07)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3161) = perspector of triangle ABC and the tangential triangle of the conic that passes through A, B, C and has center X(8)

X(3161) lies on these lines: 2,2415   6,644   8,9   37,2275   45,1213   145,1743   190,344   268,1809   329,440   355,537   404,1696

X(3161) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,8), (1222,200)
X(3161) = X(3158)-cross conjugate of X(145)
X(3161) = crosspoint of X(2) and X(145)
X(3161) = crosssum of X(649) and X(1357)

### X(3162) = X(2)-CEVA CONJUGATE OF X(25)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(2) and u : v : w = X(25)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3162) lies on these lines: 2,2138   6,66   19,614   22,112   24,1627   25,32   204,612   216,1033   232,1609   305,648   378,1180   468,1611   1194,1968   1498,1529

X(3162) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,25), (1370,455)
X(3162) = crosspoint of X(2) and X(1370)

### X(3163) = X(2)-CEVA CONJUGATE OF X(30)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(2) and u : v : w = X(30)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)
Barycentrics    (sin^2 A)(cos A - 2 cos B cos C)^2 : :

X(3163) lies on the Steiner inellipse and on these lines: 2,648   6,13   30,1990   216,549   233,547   376,577   553,1086   1084,1196   1100,1146   1636,1637   2482,2799

X(3163) = midpoint of X(2) ane X(648)
X(3163) = complement of X(1494)
X(3163) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,30), (30,3081)
X(3163) = X(3081)-cross conjugate of X(30)
X(3163) = crosspoint of X(2) and X(30)
X(3163) = crosssum of X(6) and X(74)
X(3163) = crossdifference of every pair of points on the line X(74)X(526)
X(3163) = center of circumconic that is locus of trilinear poles of lines parallel to Euler line (i.e. lines that pass through X(30))
X(3163) = perspector of circumconic centered at X(30) (parabola {A,B,C,X(30),X(476)})
X(3163) = perspector of ABC and medial triangle of cevian triangle of X(30)
X(3163) = barycentric square of X(30)
X(3163) = X(12034)-of-orthic-triangle if ABC is acute

### X(3164) = X(3)-CEVA CONJUGATE OF X(2)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(3) and u : v : w = X(2)
Barycentrics   csc 2A - csc 2B - csc 2C : csc 2B - csc 2C - csc 2A : csc 2C - csc 2A - csc 2B    (P. Moses, 5/24/07)

X(3164) lies on these lines: 2,216   6,401   20,185   22,385   69,1972   160,523   206,1632   237,3186   577,648   1976,2998   3101,3187

X(3164) = reflection of X(264) in X(216)
X(3164) = isogonal conjugate of X(1988)
X(3164) = anticomplement of X(264)
X(3164) = X(3)-Ceva conjugate of X(2)
X(3164) = anticomplementary isotomic conjugate of X(4)
X(3164) = antipedal isotomic conjugate of X(4)

### X(3165) = X(3)-CEVA CONJUGATE OF X(15)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(3) and u : v : w = X(15)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3165) lies on these lines: 15,186   16,184   35,1094   54,62   577,3165

X(3165) = X(3)-Ceva conjugate of X(15)

### X(3166) = X(3)-CEVA CONJUGATE OF X(16)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(3) and u : v : w = X(16)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3166) lies on these lines: 15,184   16,186   35,1095   54,61   577,3165

X(3166) = X(3)-Ceva conjugate of X(16)

### X(3167) = X(6)-CEVA CONJUGATE OF X(3)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(6) and u : v : w = X(3)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3167) lies on these lines: 3,49   6,1196   22,323   25,110   68,1656   154,511   193,459   195,973   392,912   542,1853   999,1201   1200,2280   1362,1397   1384,1501   1611,1692   1994,1995

X(3167) =X(i)-Ceva conjugate of X(j) for these (i,j): (6,3), (193,3053)
X(3167) = crosspoint of X(6) and X(3053)
X(3167) = crosssum of X(2) and X(2996)
X(3167) = Thomson-isogonal conjugate of X(20)
X(3167) = perspector of unary cofactor triangles of outer and inner Vecten triangles
X(3167) = centroid of anticevian triangle of X(3), which is also the antipedal triangle of X(64) and the tangential triangle of the MacBeath circumconic

### X(3168) = X(6)-CEVA CONJUGATE OF X(4)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(6) and u : v : w = X(4)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3168) lies on these lines: {2,1972}, {4,51}, {6,436}, {107,184}, {232,800}, {262,459}, {450,1993}, {1148,1870}

X(3168) = X(6)-Ceva conjugate of X(4)

### X(3169) = X(6)-CEVA CONJUGATE OF X(9)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(6) and u : v : w = X(9)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3169) lies on these lines: 1,2092   6,979   8,9   42,1449   43,2300   100,604   145,1400   284,2319   519,573   1018,1743

X(3169) = X(i)-Ceva conjugate of X(j) for these (i,j): (6,9), (3210,978)

### X(3170) = X(6)-CEVA CONJUGATE OF X(15)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(6) and u : v : w = X(15)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3170) lies on these lines: 15,323   16,184   32,3171   202,2308   1994,2005

X(3170) = X(6)-Ceva conjugate of X(15)

### X(3171) = X(6)-CEVA CONJUGATE OF X(16)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(6) and u : v : w = X(16)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3171) lies on these lines: 15,184   16,323   32,3170   61,110   203,2308   1994,2004

X(3171) = X(6)-Ceva conjugate of X(16)

### X(3172) = X(6)-CEVA CONJUGATE OF X(25)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(6) and u : v : w = X(25)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2[3a4 - 2a2b2 - 2a2c2 - (b2 - c2)2]/(b2 + c2 - a2)      (E. Danneels)

X(3172) lies on these lines: 3,112   6,64   19,1104   20,1249   24,1384   25,32   31,607   41,3195   204,3198   232,3053   608,1042   1907,3087

X(3172) = X(i)-Ceva conjugate of X(j) for these (i,j): (1249,154), (2332,1973)
X(3172) = crosspoint of X(i) and X(j) for these (i,j): (6,154), (204,3213)
X(3172) = crosssum of X(2) and X(253)

### X(3173) = X(7)-CEVA CONJUGATE OF X(3)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(7) and u : v : w = X(3)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2[b2 + c2 - a2)[a3 - a2b - a2c - a(b + c)2 + (b + c)(b2 + c2)]/(b + c - a)      (E. Danneels)

X(3173) lies on these lines: 1,90   3,1794   6,226   7,2982   55,916   63,77   109,3190   221,758   223,2323   278,651   515,1498   1362,1397   1708,2911   1936,2947   2192,2801

X(3173) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,3), (2982,1214)
X(3173) = cevapoint of X(3157) and X(3211)

### X(3174) = X(7)-CEVA CONJUGATE OF X(9)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(7) and u : v : w = X(9)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a( b + c - a)[a3 - 3a2b - 3a2c + a(3b2 - 2bc + 3c2) - (b + c)(b - c)2]      (E. Danneels)

X(3174) lies on these lines: 1,142   9,55   40,518   65,2136   78,390   100,1445   516,1490   527,2951   528,1537   936,1001   2324,2340   2801,2950

X(3174) = midpoint of X(i) and X(j) for these (i,j): (2136,3174), (2550,3183)
X(3174) = crosssum of X(663) and X(3020)
X(3174) = X(7)-Ceva conjuate of X(9)
X(3174) = X(66)-of-excentral-triangle
X(3174) = intangents-to-extangents similarity image of X(9)

### X(3175) = X(7)-CEVA CONJUGATE OF X(10)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c)(a2 + ab + ac - 2bc)     (M. Iliev, 5/13/07)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b + c)(a2 + ab + ac - 2bc)      (E. Danneels)

X(3175) lies on these lines: 2,37   72,519   190,1999   354,726   428,528   1211,2321

X(3175) = X(7)-Ceva conjugate of X(10)

### X(3176) = X(8)-CEVA CONJUGATE OF X(4)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(8) and u : v : w = X(4)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3176) lies on these lines: 1,281   4,65   8,1034   20,653   40,1712   84,1767   92,938   207,1490   243,1788   278,1210   451,3085   1863,1902

X(3176) = X(8)-Ceva conjugate of X(4)

### X(3177) = X(9)-CEVA CONJUGATE OF X(2)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(9) and u : v : w = X(2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a3(b + c) - a2(2b2 - bc + 2c2) + a(b + c)(b - c)2 - bc(b - c)2     (E. Danneels)

X(3177) lies on these lines: 2,85   63,194   105,330   144,145   220,664   329,1655   672,3212   894,2263   1654,3152

X(3177) = reflection of X(85) in X(1212)
X(3177) = anticomplement of X(85)
X(3177) = X(9)-Ceva conjugate of X(2)
X(3177) = perspector of Gemini triangle 35 and cross-triangle of Gemini triangles 35 and 37

### X(3178) = X(12)-CEVA CONJUGATE OF X(10)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(12) and u : v : w = X(10)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b + c)(a3 + 2a2b + 2a2c + abc - b3 - c3)     (E. Danneels)

X(3178) lies on this line: 1,2

X(3178) = X(12)-Ceva conjugate of X(10)

### X(3179) = X(13)-CEVA CONJUGATE OF X(1)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(13) and u : v : w = X(1)
Trilinears        tan(A/2 + π/6) : tan(B/2 + π/6) : tan(C/2 + π/6)     (M. Iliev, 4/12/07)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3179) lies on these lines: 1,15   9,46   13,484   18,1653   57,1081

X(3179) = X(13)-Ceva conjugate of X(1)
X(3179) = X(202)-cross conjugate of X(1)

### X(3180) = X(13)-CEVA CONJUGATE OF X(2)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(13) and u : v : w = X(2)

X(3180) lies on these lines: 2,6   13,533   61,634   62,619   148,531   194,617   383,1351

X(3180) = reflection of X(i) in X(j) for these (i,j): (298,396), (616,15), (621,13), (3181,385)
X(3180) = anticomplement of X(298)
X(3180) = X(13)-Ceva conjugate of X(2)
X(3180) = {X(2),X(193)}-harmonic conjugate of X(3181)
X(3180) = {X(6),X(7837)}-harmonic conjugate of X(3181)

### X(3181) = X(14)-CEVA CONJUGATE OF X(2)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(14) and u : v : w = X(2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3181) lies on these lines: 2,6   14,532   16,533   61,618   62,633   148,530   194,616   1080,1351

X(3181) = reflection of X(i) in X(j) for these (i,j): (299,395), (617,16), (622,14), (3180,385)
X(3181) = anticomplement of X(299)
X(3181) = X(14)-Ceva conjugate of X(2)
X(3181) = {X(2),X(193)}-harmonic conjugate of X(3180)
X(3181) = {X(6),X(7837)}-harmonic conjugate of X(3180)

### X(3182) = X(20)-CEVA CONJUGATE OF X(1)

Trilinears    u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(20) and u : v : w = X(1)

X(3182) lies on the Darboux cubic and these lines: 1,64   3,223   4,57   20,3347   40,3346   579,2270   658,1097   1044,1716   1490,3348   1498,3354   3183,3473

X(3182) = reflection of X(3345) in X(3)
X(3182) = X(20)-Ceva conjugate of X(1)
X(3182) = isogonal conjugate of X(3347)
X(3182) = excentral isogonal conjugate of X(7992)
X(3182) = perspector of excentral triangle and 3rd extouch triangle

### X(3183) = X(20)-CEVA CONJUGATE OF X(4)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(20) and u : v : w = X(4)

X(3183) lies on the Darboux cubic and these lines: 1,196   3,1033   4,64   19,84   20,3348   40,1712   376,1075   393,1192   1490,3354   1498,2131   1620,1990   3182,3473

X(3183) = reflection of X(3346) in X(3)
X(3183) = X(20)-Ceva conjugate of X(4)
X(3183) = crosspoint of X(20) and X(2060)
X(3183) = isogonal conjugate of X(3348)
X(3183) = orthocenter of cevian triangle of X(20)
X(3183) = perspector of hexyl triangle and anticevian triangle of X(1712)

### X(3184) = X(20)-CEVA CONJUGATE OF X(30)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(20) and u : v : w = X(30)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3184) lies on these lines: 3,113   20,107   30,133   112,376

X(3184) = inverse-in-circumcircle of X(2935)
X(3184) = midpoint of X(20) and X(107)
X(3184) = reflection of X(122) in X(3)
X(3184) = X(20)-Ceva conjugate of X(30)
X(3184) = center of conic {{A,B,C,X(20),X(107)}}

### X(3185) = X(21)-CEVA CONJUGATE OF X(6)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(21) and u : v : w = X(6)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a3[a2b + a2c - abc - b3 - c3]     (E. Danneels)
X(3185) lies on these lines: 1,859   3,960   12,1894   19,25   21,1610   31,48   35,2933   47,1437   100,312   184,2361   212,692   227,1875   674,3190   851,1836   1011,2182   1460,2178   1630,2328   1953,1962   2176,2223   2179,2304   2308,2317

X(3185) = isogonal conjugate of X(2995)
X(3185) = X(21)-Ceva conjugate of X(6)
X(3185) = crosspoint of X(100) and X(2149)
X(3185) = crosssum of X(123) and X(525)
X(3185) = crossdifference of every pair of points on the line X(905)X(1577)

### X(3186) = X(25)-CEVA CONJUGATE OF X(4)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b2c2 - a2b2 - a2c2)/(b2 + c2 - a2)     (M. Iliev, 5/13/07)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (a2b2 + a2c2 - b2c2)/(b2 + c2 - a2)     (E. Danneels)

X(3186) lies on these lines: {1,242}, {2,3505}, {4,69}, {6,419}, {19,2319}, {24,1075}, {25,385}, {141,5117}, {232,800}, {237,3164}, {393,694}, {427,3314}, {571,1632}, {648,1974}, {1714,3144}, {3462,3542}

X(3186) = isogonal conjugate of X(3504)
X(3186) = X(25)-Ceva conjugate of X(4)
X(3186) = X(1613)-cross conjugate of X(194)
X(3186) = crossdifference of every pair of points on the line X(2524)X(3049)
X(3186) = pole wrt polar circle of trilinear polar of X(2998) (line X(512)X(625))
X(3186) = polar conjugate of X(2998)

### X(3187) = X(27)-CEVA CONJUGATE OF X(2)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(a3 + a2b + a2c - b2c - bc2)     (M. Iliev, 5/13/07)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3187) lies on these lines: 1,2   7,1943   31,740   75,81   92,1172   100,2352   193,1839   335,2606   1724,2901   3101,3164

X(3187) = anticomplement of X(306)
X(3187) = X(27)-Ceva conjugate of X(2)
X(3187) = crosspoint of X(648) and X(1016)
X(3187) = crosssum of X(647) and X(1015)
X(3187) = crossdifference of every pair of points on the line X(649)X(838)

### X(3188) = X(27)-CEVA CONJUGATE OF X(7)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(27) and u : v : w = X(7)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = [a5 - a4(b + c) - a3(b2 + c2) + a2(b3 + c3) + bc(b + c)(b - c)2]/(b + c - a)     (E. Danneels)

X(3188) lies on these lines: 1,7   3,1446   21,85   28,272   241,379   348,377   514,1729   917,934   958,1441

X(3188) = X(27)-Ceva conjugate of X(7)

### X(3189) = X(27)-CEVA CONJUGATE OF X(9)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(27) and u : v : w = X(9)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b + c - a)[3a3 + a2(b + c) - a(b + c)2 + (b + c)(b - c)2]     (E. Danneels)

X(3189) lies on these lines: 1,142   4,2900   8,21   20,518   40,476   65,145   71,3169   78,497   100,1788   200,950   210,452   281,2332   346,2264   380,2321   390,960   528,962   938,1376   997,1058   1118,1897

X(3189) = reflection of X(2550) in X(3174)
X(3189) = X(27)-Ceva conjugate of X(9)
X(3189) = crosssum of X(1357) and X(1459)
X(3189) = intangents-to-extangents similarity image of X(8)

### X(3190) = X(29)-CEVA CONJUGATE OF X(9)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(29) and u : v : w = X(9)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3190) lies on these lines: 1,2   6,1260   9,2318   25,101   33,2900   57,1818   58,1259   63,991   72,581   100,1754   109,3173   197,1630   209,579   212,2323   219,284   228,573   318,2901   516,2947   517,3198   518,1214   674,3185   1331,1993   1612,1724   1897,2052   2172,2187

X(3190) = X(29)-Ceva conjugate of X(9)
X(3190) = crosspoint of X(1252) and X(1897)
X(3190) = crosssum of X(1086) and X(1459)
X(3190) = crossdifference of every pair of points on the line X(649)X(2504)

### X(3191) = X(29)-CEVA CONJUGATE OF X(10)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(29) and u : v : w = X(10)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3191) lies on these lines: 1,6   28,101   33,2901   40,228   201,758   329,581   568,1153   943,2328   1331,3193   1824,2910   1896,1897

X(3191) = X(29)-Ceva conjugate of X(10)

### X(3192) = X(29)-CEVA CONJUGATE OF X(19)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(29) and u : v : w = X(19)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3192) lies on these lines: 1,406   4,386   6,25   19,2258   24,58   31,1182   33,42   34,1193   53,430   55,3195   73,208   162,2185   199,577   216,1011   235,1834   387,3089   461,1249   475,3216   581,3194   995,1870   2207,2271

X(3192) = X(29)-Ceva conjugate of X(19)

### X(3193) = X(29)-CEVA CONJUGATE OF X(21)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(29) and u : v : w = X(21)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3193) lies on these lines: 1,21   4,155   28,110   29,1069   40,1790   46,453   57,1819   225,651   323,2475   377,394   517,1437   648,1896   1068,3157   1331,3191   1442,1444   1816,1936   2287,2323

X(3193) = X(29)-Ceva conjugate of X(21)
X(3193) = cevapoint of X(i) and X(j) for these (i,j): (1,155), (46,3157)
X(3193) = X(i)-cross conjugate of X(j) for these (i,j): (1800,21), (3157,1800)
X(3193) = crosspoint of X(1) and X(155) wrt both the excentral and tangential triangles

### X(3194) = X(29)-CEVA CONJUGATE OF X(28)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(29) and u : v : w = X(28)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3194) lies on these lines: 1,204   4,6   28,34   29,81   40,2331   73,108   112,972   196,221   223,2360   240,1046   329,3195   580,1715   581,3192   937,1474   1104,1870   1193,1430   1203,1838   1741,1778   1828,2203   1875,2194   2266,2332

X(3194) = X(i)-Ceva conjugate of X(j) for these (i,j): (29,28), (81,1172)
X(3194) = cevapoint of X(i) and X(j) for these (i,j): (6,204), (208,221), (2331,3195)
X(3194) = X(221)-cross conjugate of X(2360)

### X(3195) = X(33)-CEVA CONJUGATE OF X(25)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(33) and u : v : w = X(25)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3195) lies on these lines: 6,33   25,31   41,3172   55,3192   108,222   154,478   208,221   213,607   329,3194   611,1957   614,1876   1193,1593   1201,1398   2187,3209

X(3195) = X(i)-Ceva conjugate of X(j) for these (i,j): (6,607), (33,25), (208,3209), (3194,2331)
X(3195) = crosspoint of X(i) and X(j) for these (i,j): (6,221), (208,2331)
X(3195) = crosssum of X(i) and X(j) for these (i,j): (2,280), (905,2968)

### X(3196) = X(36)-CEVA CONJUGATE OF X(55)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(36) and u : v : w = X(55)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3196) lies on these lines: 6,101   9,1030   36,44   45,55   198,2265   909,3207

X(3196) = isogonal conjugate of X(8046)
X(3196) = X(i)-Ceva conjugate of X(j) for these (i,j): (36,55), (44,6)
X(3196) = crosssum of X(650) and X(3025)
X(3196) = crossdifference of every pair of poins on the line X(900)X(1387)

### X(3197) = X(40)-CEVA CONJUGATE OF X(55)

Barycentrics   a^2*(a^6 - 2*a^5*b - a^4*b^2 + 4*a^3*b^3 - a^2*b^4 - 2*a*b^5 + b^6 - 2*a^5*c - 2*a^4*b*c + 2*a*b^4*c + 2*b^5*c - a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 + 4*a^3*c^3 - 4*b^3*c^3 - a^2*c^4 + 2*a*b*c^4 - b^2*c^4 - 2*a*c^5 + 2*b*c^5 + c^6) : :
X(3197) = 3 X[154] - 2 X[18621]

X(3197) lies on the cubics K179 and K750, and on these lines:
{3,1630}, {6,19}, {9,6001}, {22,11445}, {24,11460}, {25,3611}, {37,1854}, {40,219}, {48,55}, {64,71}, {101,1604}, {109,15905}, {155,8141}, {159,20468}, {184,11406}, {218,2270}, {222,18725}, {268,12330}, {281,5776}, {394,3101}, {517,3211}, {910,2911}, {958,15656}, {966,20306}, {1030,3207}, {1181,6197}, {1190,2266}, {1191,21770}, {1376,10174}, {1407,26934}, {1503,2550}, {1615,7964}, {1802,12335}, {1853,3925}, {1903,7079}, {1993,9536}, {2093,18594}, {2183,7355}, {2259,11051}, {2289,10310}, {2301,4254}, {2343,8602}, {3196,5036}, {3198,3990}, {3779,9924}, {4258,11434}, {5415,17819}, {5416,17820}, {5928,30686}, {6237,17834}, {6252,17840}, {6253,17845}, {6255,17846}, {6404,17843}, {6759,10306}, {6769,22153}, {7291,23144}, {7688,10606}, {7719,9119}, {7724,17835}, {7957,7973}, {8251,17814}, {8539,17813}, {8802,17833}, {8804,12779}, {9537,11441}, {9816,17825}, {10119,17847}, {10319,17811}, {10636,17826}, {10637,17827}, {10902,17821}, {11206,17784}, {11428,17809}, {11435,17810}, {11471,15811}, {12417,17836}, {12661,17838}, {12662,17839}, {12663,17842}, {13041,17841}, {13042,17844}, {13567,18921}, {15509,24310}, {17837,22840}, {18405,18406}, {18451,18453}, {18598,22132}, {19132,19133}, {19180,19181}, {19430,19432}, {19431,19433}, {21778,21779}, {26908,26909}, {26952,26953}, {26957,26958}

X(3197) = isogonal conjugate of the polar conjugate of X(3176)
X(3197) = X(i)-Ceva conjugate of X(j) for these (i,j): {40, 55}, {219, 6}, {610, 198}
X(3197) = X(i)-isoconjugate of X(j) for these (i,j): {2, 3345}, {57, 1034}, {63, 7149}, {75, 7152}, {81, 8806}, {85, 7037}, {189, 3342}, {333, 8811}, {348, 7007}, {8064, 17896}
X(3197) = crosspoint of X(100) and X(23984)
X(3197) = crosssum of X(i) and X(j) for these (i,j): {9, 6769}, {514, 16596}, {650, 3318}
X(3197) = crossdifference of every pair of points on line {521, 14837}
X(3197) = barycentric product X(i)*X(j) for these {i,j}: {1, 1490}, {3, 3176}, {8, 1035}, {40, 3341}, {55, 5932}, {65, 13614}, {72, 8885}, {78, 207}, {109, 14302}
X(3197) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 7149}, {31, 3345}, {32, 7152}, {42, 8806}, {55, 1034}, {207, 273}, {1035, 7}, {1402, 8811}, {1490, 75}, {2175, 7037}, {2187, 3342}, {2212, 7007}, {3176, 264}, {3341, 309}, {5932, 6063}, {8885, 286}, {13614, 314}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 21767, 221}, {19, 2261, 2264}, {19, 19350, 6}, {48, 2272, 1436}, {55, 10536, 154}, {607, 1409, 6}, {610, 1498, 7152}, {2187, 2357, 55}, {5584, 6254, 64}, {10536, 11190, 55}

### X(3198) = X(40)-CEVA CONJUGATE OF X(71)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(40) and u : v : w = X(71)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3198) lies on these lines: 10,440   12,1869   19,25   31,2264   40,64   42,65   100,1297   204,3172   210,1903   212,2182   517,3190   1011,1212   1071,1715   1108,2352   1260,1766

X(3198) = X(i)-Ceva conjugate of X(j) for these (i,j): (40,71), (72,37), (200,42)
X(3198) = crosspoint of X(20) and X(610)
X(3198) = crosssum of X(64) and X(2184)

### X(3199) = X(53)-CEVA CONJUGATE OF X(51)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(53) and u : v : w = X(51)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3199) lies on these lines: 4,39   5,53   6,1598   24,187   25,32   33,1500   34,1015   51,217   52,1626   97,2984   115,235   297,626   393,800   427,1506   574,1593   1504,3092   1505,3093   1574,1861   1692,1974   1843,2211

X(3199) = X(53)-Ceva conjugate of X(51)
X(3199) = crosspoint of X(25) and X(393)
X(3199) = crosssum of X(69) and X(394)

### X(3200) = X(54)-CEVA CONJUGATE OF X(15)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(54) and u : v : w = X(15)
Trilinears        sin(A + π/3) sin(2A + 2π/3) : sin(B + π/3) sin(2B + 2π/3) : sin(C + π/3) sin(2C + 2π/3)     (M. Iliev, 4/12/07)
Trilinears        cos 3A + cos(A + π/3) : cos 3B + cos(B + π/3) : cos 3C + cos(C + π/3)     (M. Iliev, 4/12/07)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3200) lies on these lines: 3,3205   6,3201   13,110   16,184   18,54   49,62   61,1147   202,215   3105,3202

X(3200) = X(54)-Ceva conjugate of X(15)
X(3200) = crosspoint of X(15) and X(62)
X(3200) = crosssum of X(13) and X(18)

### X(3201) = X(54)-CEVA CONJUGATE OF X(16)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(54) and u : v : w = X(16)

Trilinears        sin(A - π/3) sin(2A - 2π/3) : sin(B - π/3) sin(2B - 2π/3) : sin(C - π/3) sin(2C - 2π/3)     (M. Iliev, 4/12/07)
Trilinears        cos 3A + cos(A - π/3) : cos 3B + cos(B - π/3) : cos 3C + cos(C - π/3)     (M. Iliev, 4/12/07)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3201) lies on these lines: 3,3206   6,3200   14,110   15,184   17,54   49,61   62,1147   203,215   3104,3202

X(3201) = X(i)-Ceva conjugate of X(j) for these (i,j): (54,16), (2981,50)
X(3201) = crosspoint of X(16) and X(61)
X(3201) = crosssum of X(14) and X(17)

### X(3202) = X(54)-CEVA CONJUGATE OF X(32)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(54) and u : v : w = X(32)
Trilinears        g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a5[a2(b2 + c2) - b4 - b2c2 - c4]      (E. Danneels)

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). X(3202) = X(39)-of-A'B'C'. (Randy Hutson, January 15, 2019)

X(3202) lies on these lines: 6,3203   26,206   39,184   49,3095   76,110   156,2782   1916,3044   3104,3201   3105,3200   3106,3206   3107,3205

X(3202) = X(54)-Ceva conjugate of X(32)
X(3202) = crosspoint of X(1670) and X(1671)
X(3202) = crosssum of X(1676) and X(1677)

### X(3203) = X(54)-CEVA CONJUGATE OF X(39)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(54) and u : v : w = X(39)
Trilinears        g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a3(b2 + c2)(a4 - b2c2 - c2a2 - a2b2)      (E. Danneels)
Barycentrics   ag(a,b,c) : bg(b,c,a) : cg(c,a,b)

X(3203) lies on these lines: 6,3202   32,184   83,110   140,141

X(3203) = X(i)-Ceva conjugate of X(j) for these (i,j): (54,39), (110,3050)
X(3203) = crosspoint of X(1342) and X(1343)

### X(3204) = X(54)-CEVA CONJUGATE OF X(55)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(54) and u : v : w = X(55)
Trilinears        g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a[a3 - a(b + c)2 + bc(b + c)]      (E. Danneels)
Barycentrics   ag(a,b,c) : bg(b,c,a) : cg(c,a,b)

X(3204) lies on these lines: 6,101   9,2174   41,584   45,284   218,583   220,2301   1953,2246   2176,2220

X(3204) = X(54)-Ceva conjugate of X(55)

### X(3205) = X(54)-CEVA CONJUGATE OF X(61)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(54) and u : v : w = X(61)
Trilinears        sin(A + π/6) sin(2A - π/6) : sin(B + π/6) sin(2B - π/6) : sin(C + π/6) sin(2C - π/6)     (M. Iliev, 4/12/07)
Trilinears        cos 3A - cos(A - π/3) : cos 3B - cos(B - π/3) : cos 3C - cos(C - π/3)     (M. Iliev, 4/12/07)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3205) lies on these lines: 3,3200   6,3206   13,156   14,54   15,1147   17,110   49,61   62,184   203,2477   3107,3202

X(3205) = X(54)-Ceva conjugate of X(61)

### X(3206) = X(54)-CEVA CONJUGATE OF X(62)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(54) and u : v : w = X(62)
Trilinears        sin(A - π/6) sin(2A + π/6) : sin(B - π/6) sin(2B + π/6) : sin(C - π/6) sin(2C + π/6)     (M. Iliev, 4/12/07)
Trilinears        cos 3A - cos(A + π/3) : cos 3B - cos(B + π/3) : cos 3C - cos(C + π/3)     (M. Iliev, 4/12/07)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)

X(3206) lies on these lines: 3,3201   6,3205   13,54   14,156   16,1147   18,110   49,62   61,184   202,2477   3106,3202

X(3206) = X(54)-Ceva conjugate of X(62)

### X(3207) = X(55)-CEVA CONJUGATE OF X(6)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(55) and u : v : w = X(6)
Trilinears    a[3a2 - 2ab - 2ac - (b - c)2] : :      (E. Danneels)

X(3207) lies on these lines: 1,910   3,101   6,41   31,1191   36,218   45,2182   169,1385   595,1384   909,3196   944,1146   1030,3197   1100,2270   1190,1617   1319,2082   1334,2272   1376,2329   1388,2170   1604,1630   1616,1914   1696,2268   2053,2110   2176,2223   2242,2271

X(3207) = X(55)-Ceva conjugate of X(6)
X(3207) = crosspoint of X(i) and X(j) for these (i,j): (101,1262), (165,1419)
X(3207) = crosssum of X(514) and X1146)
X(3207) = trilinear pole wrt tangential triangle of antiorthic axis

### X(3208) = X(55)-CEVA CONJUGATE OF X(9)

Trilinears    (b + c - a)(bc - ab - ac) : :    (M. Iliev, 5/13/2007)
Trilinears    (b + c) (a - b) (a - c) - a b c : :

X(3208) lies on these lines: 1,39   8,9   43,2176   145,672   192,1423   2340,3158   3057,3061

X(3208) = X(i)-Ceva conjugate of X(j) for these (i,j): (55,9), (192,43), (196,221), (208,3195)
X(3208) = {X(1),X(1018)}-harmonic conjugate of X(3501)

### X(3209) = X(56)-CEVA CONJUGATE OF X(25)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(56) and u : v : w = X(25)
Trilinears        g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a[a3 + a2(b + c) - a( b + c)2 - (b + c)(b - c)2]/[(b + c - a)(b2 + c2 - a2)]      (E. Danneels)
Barycentrics   ag(a,b,c) : bg(b,c,a) : cg(c,a,b)

X(3209) lies on these lines: 19,56   25,1096   108,281   196,347   198,208   604,608   607,1400   1593,2285   2187,3195

X(3209) = X(i)-Ceva conjugate of X(j) for these (i,j): (19,608), (56,25), (196,221), (208,3195)
X(3209) = crosspoint of X(19) and X(2331)
X(3209) = X(112)-beth conjugate of X(6)

### X(3210) = X(57)-CEVA CONJUGATE OF X(2)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(57) and u : v : w = X(2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2(b + c) + a(b2 - bc + c2) - bc(b + c)      (E. Danneels)

X(3210) lies on these lines: 2,37   8,38   43,726   57,1999   63,194   81,330   100,1403   226,1266   384,2221   664,1407   740,982   3011,3021   3101,3164

X(3210) = anticomplement of X(312)
X(3210) = X(i)-Ceva conjugate of X(j) for these (i,j): (57,2), (978,3169)
X(3210) = crosssum of X(663) and X(1977)
X(3210) = polar conjugate of isogonal conjugate of X(20805)
X(3210) = perspector of Gemini triangle 36 and cross-triangle of Gemini triangles 36 and 38

### X(3211) = X(57)-CEVA CONJUGATE OF X(3)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(57) and u : v : w = X(3)
Trilinears        g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a(b2 + c2 - a2)[a4 - 2a3(b + c) - 2a2bc + 2a(b3 + c3) - (b2 - c2)2]      (E. Danneels)
Barycentrics   ag(a,b,c) : bg(b,c,a) : cg(c,a,b)

X(3211) lies on these lines: 3,48   6,169   57,3173   155,610   579,2178   651,1119   1069,2164   1200,2280   1467,1743

X(3211) = X(i)-Ceva conjugate of X(j) for these (i,j): (57,3), (3173,3157)
X(3211) = crosssum of X(650) and X(2969)

### X(3212) = X(57)-CEVA CONJUGATE OF X(7)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(bc - ab - ac)/(b + c - a)     (M. Iliev, 5/13/07)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3212) lies on these lines: 1,1447   2,257   7,8   57,239   241,2275   273,1829   279,291   348,1788   607,653   672,3177   1400,2998   1431,1966   1445,2082

X(3212) = isogonal conjugate of X(2053)
X(3212) = anticomplement of X(3061)
X(3212) = X(57)-Ceva conjugate of X(7)
X(3212) = cevapoint of X(43) and X(1423)
X(3212) = X(43)-cross conjugate of X(192)
X(3212) = crosspoint of X(57) and X(1423)
X(3212) = crosssum of X(9) and X(2319)

### X(3213) = X(57)-CEVA CONJUGATE OF X(34)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(57) and u : v : w = X(34)
Trilinears        g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = [3a4 - 2a2(b2 + c2) - (b2 - c2)2]/[(b + c - a)(b2 + c2 - a2)]      (E. Danneels)
Barycentrics   ag(a,b,c) : bg(b,c,a) : cg(c,a,b)

X(3213) lies on these lines: 6,208   19,56   33,2285   34,604   48,2331   57,1172   154,204   196,380   608,1407   610,1249

X(3213) = X(i)-Ceva conjugate of X(j) for these (i,j): (57,34), (1172,608)
X(3213) = X(3172)-cross conjugate of X(204)
X(3213) = crosspoint of X(57) and X(1394)
X(3213) = X(112)-beth conjugate of X(48)

### X(3214) = X(57)-CEVA CONJUGATE OF X(37)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(a2 + ab + ac - 2bc)     (M. Iliev, 5/13/07)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3214) lies on these lines: 1,2   44,71   65,2137   209,2390   210,2292   405,2177   902,1724   1334,2238   1376,1468   1458,1788   1475,1575   1834,2318   2246,2333

X(3214) = reflection of X(1201) in X(3216)
X(3214) = X(57)-Ceva conjugate of X(37)

### X(3215) = X(57)-CEVA CONJUGATE OF X(48)

Trilinears        u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(57) and u : v : w = X(48)
Trilinears        g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2(b2 + c2 - a2)[a3 - a2(b + c) - a(b + c)2 + (b + c)(b2 + c2)]/(b + c - a)      (E. Danneels)
Barycentrics   ag(a,b,c) : bg(b,c,a) : cg(c,a,b)

X(3215) lies on these lines: 3,73   6,1195   31,65   34,46   41,2253   221,2361   224,1331   225,1754   283,1038   377,1935   582,1465   602,1457   1496,2646   1708,1780   1724,1877   1726,1825   2199,2245

X(3215) = X(57)-Ceva conjugate of X(48)

### X(3216) = X(58)-CEVA CONJUGATE OF X(1)

Trilinears    u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z = X(58) and u : v : w = X(1)
Trilinears    a2(b + c) + a(b2 + c2) - bc(b + c)      (E. Danneels)

Let A'B'C' be the medial triangle and A''B''C'' then cevian triangle of X(1). Let U be the circumcircle of A, A', A'', and define V and W cyclically. Then X(3216) is the radical center of U, V, W. See Antreas Hatzipolakis and Peter Moses, Hyacinthos 267292.

X(3216) lies on on the Feuerbach of the tangential triangle and these lines: {1, 2}, {3, 1724}, {4, 5400}, {6, 474}, {9, 4261}, {21, 4256}, {35, 238}, {36, 5247}, {38, 3678}, {39, 2238}, {44, 3916}, {46, 2390}, {56, 4551}, {58, 404}, {65, 1739}, {72, 3670}, {73, 3911}, {100, 595}, {140, 5396}, {155, 6911}, {171, 1203}, {213, 1575}, {216, 1713}, {244, 3874}, {267, 2640}, {284, 7523}, {392, 4646}, {405, 4255}, {411, 1780}, {475, 3192}, {500, 549}, {517, 3987}, {518, 3953}, {579, 610}, {580, 6905}, {581, 631}, {602, 6796}, {662, 849}, {748, 5248}, {872, 6533}, {960, 4424}, {966, 5105}, {970, 1764}, {979, 2163}, {982, 5904}, {986, 5692}, {991, 3523}, {992, 2092}, {1015, 3780}, {1018, 2176}, {1045, 5506}, {1046, 1054}, {1050, 5541}, {1064, 6684}, {1104, 5440}, {1191, 5687}, {1213, 5153}, {1376, 5264}, {1450, 10106}, {1453, 5438}, {1457, 4848}, {1491, 4040}, {1498, 1754}, {1574, 2295}, {1654, 13571}, {1730, 10974}, {1738, 12047}, {1740, 2228}, {1757, 4283}, {1788, 10571}, {1834, 4187}, {1964, 4974}, {2077, 3073}, {2108, 6196}, {2234, 4672}, {2276, 3294}, {2292, 10176}, {2594, 5433}, {2650, 5883}, {2901, 4358}, {2915, 2916}, {2940, 2959}, {3191, 3772}, {3264, 3875}, {3290, 3970}, {3338, 3751}, {3454, 4202}, {3555, 4694}, {3570, 7760}, {3666, 5044}, {3684, 5299}, {3725, 4647}, {3740, 4719}, {3876, 4850}, {3878, 4642}, {3915, 8715}, {4003, 4005}, {4188, 4257}, {4205, 5241}, {4210, 4278}, {4300, 10164}, {4306, 5435}, {4413, 5711}, {4507, 9433}, {4653, 5047}, {4674, 5903}, {5255, 5315}, {5398, 6924}, {5706, 6918}, {5710, 9709}, {5713, 6854}, {5718, 8728}, {5721, 6922}, {9363, 13370}, {9548, 10882}, {9549, 12435}, {9567, 10441}

X(3216) = midpoint of X(1201) and X(3214)
X(3216) = reflection of X(1) in X(1201)
X(3216) = X(i)-Ceva conjugate of X(j) for these (i,j): {58, 1}, {404, 3}
X(3216) = X(3159)-cross conjugate of X(1)
X(3216) = crosspoint of X(662) and X(7035)
X(3216) = crossdifference of every pair of points on line (649, 3726)
X(3216) = crosssum of X(i) and X(j) for these (i,j): {513, 8054}, {661, 3248}
X(3216) = X(i)-aleph conjugate of X(j) for these (i,j): {1, 191}, {6, 1045}, {21, 20}, {28, 1714}, {58, 3216}, {81, 2}, {100, 4427}, {174, 1762}, {259, 2938}, {266, 1046}, {365, 846}, {366, 1761}, {509, 1781}, {662, 3882}, {6727, 3}
X(3216) = X(3699)-beth conjugate of X(3216)
X(3216) = X(741)-he conjugate of X(6)
X(3216) = X(i)-zayin conjugate of X(j) for these (i,j): {56, 1724}, {58, 3216}, {667, 4040}, {1193, 1}, {1203, 3293}, {2260, 1743}, {2308, 43}, {2309, 87}, {3122, 9359}, {3733, 3737}, {4057, 513}, {10457, 58}
X(3216) = barycentric product X(81)*X(3159)
X(3216) = barycentric quotient X(3159)/X(321)
X(3216) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 43, 3293), (1, 6048, 3679), (2, 386, 1), (2, 9534, 10479), (3, 4383, 1724), (8, 995, 1), (10, 1193, 1), (42, 1125, 1), (43, 978, 1), (72, 3752, 3670), (614, 3811, 1), (899, 1193, 10), (936, 2999, 1), (975, 5256, 1), (1046, 1054, 3336), (1149, 3244, 1), (1698, 5313, 1), (3624, 5312, 1), (3751, 11512, 3338), (4202, 5741, 3454), (9534, 10479, 3679)

### X(3217) = X(58)-CEVA CONJUGATE OF X(55)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(a2 + ab + ac - 2bc)     (M. Iliev, 5/13/07)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3217) lies on these lines: 6,1201   9,21   19,1830   37,2280   44,48   101,604   169,2171   198,672   218,1400   219,2347   220,2269   391,2329   644,3169   1802,2264   2174,2267   2183,2911

X(3217) = X(i)-Ceva conjugate of X(j) for these (i,j): (58,55), (983,1253)
X(3217) = crosssum of X(514) and X(3020)

### X(3218) =  INTERSECTION OF LINES X(2)X(7) AND X(8)X(46)

Trilinears    a2 - b2 - c2 + bc : :
Trilinears    (1 - 2 cos A) csc A : :     (M. Iliev, 4/12/07)

This point occurs in a Sadi Abu-Saymeh, Mowaffaq Hajja, and Hellmuth Stachel, "Another cubic associated with the triangle," Journal for Geometry and Graphics 11 (2007) 15-26.

Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the antiorthic axis. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A"B"C" be the reflection of A'B'C' in the antiorthic axis. The triangle A"B"C" is homothetic to ABC, with center of homothety X(908) and centroid X(3218); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)

X(3218) lies on these lines: 1,89   2,7   8,46   11,1776   21,942   22,1473   31,982   36,214   37,2666   38,171   40,145   44,88   65,2975   72,404   75,1150   80,535   81,593   97,1214   100,518   104,517   110,2651   149,516   153,1512   189,2994   191,1125   222,1993   229,1098   238,244   239,514   240,1430   241,1252   278,1748   291,2239   320,2245   323,1443   324,1947   333,2160   335,675   354,1621   388,1454   394,1407   411,1071   484,519   614,1707   651,1465   750,984   799,1921   899,1054   903,2161   919,1814   962,1158   986,1468   990,2000   1012,2095   1046,1193

X(3218) = reflection of X(i) in X(j) for these (i,j): (100,1155), (153,1512), (3257,44)
X(3218) = isogonal conjugate of X(2161)
X(3218) = isotomic conjugate of X(18359)
X(3218) = complement of X(17484)
X(3218) = anticomplement of X(908)
X(3218) = X(903)-Ceva conjugate of X(1)
X(3218) = cevapoint of X(36) and X(2323)
X(3218) = X(i)-cross conjugate of X(j) for these (i,j): (36,1443), (2245,36)
X(3218) = crosspoint of X(81) and X(88)
X(3218) = crosssum of X(i) and X(j) for these (i,j): (37,44), (649,2087), (1635,2170)
X(3218) = crossdifference of every pair of points on the line X(42)X(663)
X(3218) = inverse-in-circumconic-centered-at-X(9) of X(2)
X(3218) = {X(9),X(57)}-harmonic conjugate of X(3306)
X(3218) = X(1495)-of-excentral-triangle
X(3218) = endo-homothetic center of X(3)- and X(4)-Ehrmann triangles

### X(3219) = INTERSECTION OF LINES X(2)X(7) AND X(8)X(90)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2 - b2 - c2 - bc
Trilinears        (1 + 2 cos A) csc A : (1 + 2 cos B) csc B : (1 + 2 cos C) csc C     (M. Iliev, 4/12/07)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

This point occurs in a forthcoming paper by Sadi Abu-Saymeh, Mowaffaq Hajja, and Hellmuth Stachel, "Another cubic associated with the triangle."

X(3219) lies on these lines: 1,2308   2,7   8,90   10,191   21,72   31,984   37,81   38,238   42,846   45,940   55,1776   71,1654   100,210   101,1790   172,593   190,321   201,1935   219,1993   220,394   281,1748   306,2895   312,1150   323,1442   324,1948   612,1707   651,1214   748,982   756,896   799,1920   912,1006   938,1728   958,2099   960,1319   1082,2307   1268,2160

X(3219) = isogonal conjugate of X(2160)
X(3219) = isotomic conjugate of X(30690)
X(3219) = X(1268)-Ceva conjugate of X(1)
X(3219) = cevapoint of X(9) and X(191)
X(3219) = X(35)-cross conjugate of X(1442)
X(3219) = crosssum of X(i) and X(j) for these (i,j): (649,3125), (1652,1653)
X(3219) = crossdifference of every pair of points on the line X(663)X(2520)
X(3219) = {X(9),X(57)}-harmonic conjugate of X(3305)

### X(3220) = INTERSECTION OF LINES X(3)X(9) AND X(36)X(238)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[a4 - b4 - c4 + bc(b2 + c2 - a2)]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3220) is the homothetic center of the following two triangles: tangential triangles of the excentral triangle, and excentral triangle of the tangential triangle. (Note from Peter Moses, 1/24/2007)

X(3220) lies on these lines: 1,159   3,9   19,990   22,63   25,57   35,984   36,238   48,991   56,269   58,1474   100,2751   101,1818   103,2272   104,2728   154,222   165,197   184,2003   219,1350   511,2323   603,2212   759,2722   1040,1763

X(3220) = crosspoint of X(58) and X(103)
X(3220) = crosssum of X(i) and X(j) for these (i,j): (10,516), (71,2340)
X(3220) = crossdifference of every pair of points on the line X(37)X(2509)

### X(3221) = SS(a → bc) OF X(647)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 - c2)(c2a2 + a2b2 - b2c2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Trilinears for X(3221) are obtained by applying the symbolic substitution (a,b,c) → (bc,ca,ab) to trilinear coordinates of X(647). As a symbolic substitution, this mapping takes lines onto lines. In particular, points on the Euler line, X(2)X(3), with coefficients given by X(647), are mapped to the line X(6)X(194), with coefficients given by X(3221). Symbolic substitutions are introduced in the following article:

C. Kimberling, "Symbolic substitutions in the transfigured plane of a triangle," Aequationes Mathematicae 73 (2007) 156-171.

As the isogonal conjugate of a point on the circumcircle, X(3221) lies on the line at infinity.

X(3221) lies on these (parallel) lines: 30,511   669,2451   882,1843

X(3221) = isogonal conjugate of X(3222)
X(3221) = perspector of hyperbola {{A,B,C,X(2),X(194)}}
X(3221) = bicentric difference of PU(154)
X(3221) = ideal point of PU(154)

### X(3222) = ISOGONAL CONJUGATE OF X(3221)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a(b2 - c2)(c2a2 + a2b2 - b2c2)]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3222) lies on the circumcircle and these lines: 2,699   111,2998   645,803   729,1078   733,1799   799,932

X(3222) = isogonal conjugate of X(3221)
X(3222) = cevapoint of X(2) and X(669)
X(3222) = X(670)-cross conjugate of X(99)
X(3222) = trilinear pole of line X(6)X(194)
X(3222) = Ψ(X(6), X(194))
X(3222) = circumcircle intercept, other than A, B, C, of conic {{A,B,C,PU(148)}}

### X(3223) = ISOGONAL CONJUGATE OF X(1740)

Trilinears        1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) : f(b,c,a) : f(c,a,b) = X(1740)
Barycentrics   a/f(a,b,c) : b/f(b,c,a) : c/f(c,a,b)

X(3223) lies on these lines: 1,1965   31,1582   42,192   43,213   63,1967   1045,2258

X(3223) = isogonal conjugate of X(1740)
X(3223) = X(75)-cross conjugate of X(1)

### X(3224) = ISOGONAL CONJUGATE OF X(194)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(a2b2 + a2c2 - b2c2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3224) lies on these lines: 6,194   32,1613   43,213   419,2207   729,1078

X(3224) = isogonal conjugate of X(194)
X(3224) = cevapoint of X(2) and X(2998)
X(3224) = X(2)-cross conjugate of X(6)
X(3224) = isotomic conjugate of X(6374)
X(3224) = trilinear pole of line X(669)X(2451)
X(3224) = barycentric product of PU(154)

### X(3225) = SS(a → bc) OF X(98)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc/(a2b4 + a2c4 - b2c4 - c 2b4)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Trilinears for X(3225) are obtained by applying the symbolic substitution (a,b,c) → (bc,ca,ab) to trilinear coordinates of X(98). As a symbolic substitution, this mapping takes circumconics to circumconics. In particular, it maps circumcircle onto the Steiner circumellipse, which passes through X(3225).

X(3225) lies on the Steiner circumellipse and these lines: 6,670   32,99   190,1918   290,2422   648,1974   729,886

X(3225) = isogonal conjugate of X(3229)
X(3225) = isotomic conjugate of X(698)
X(3225) = cevapoint of X(i) and X(j) for these (i,j): (6,385), (192,2238)
X(3225) = X(694)-cross conjugate of X(98)
X(3225) = trilinear pole of PU(148) (line X(2)X(669))

### X(3226) = SS(a → bc) OF X(105)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc/(ab2 + ac2 - bc2 - b2c)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Trilinears for X(3226) are obtained by applying the symbolic substitution (a,b,c) → (bc,ca,ab) to trilinear coordinates of X(105). As a symbolic substitution, this mapping takes circumconics to circumconics. In particular, it maps circumcircle onto the Steiner circumellipse, which passes through X(3226).

X(3226) lies on the Steiner circumellipse and these lines: 1,668   6,190   56,664   58,99   75,87   86,670   239,292   648,1474   666,1438   1027,2481

X(3226) = isogonal conjugate of X(3009)
X(3226) = isotomic conjugate of X(726)
X(3226) = cevapoint of X(1) and X(329)
X(3226) = X(i)-cross conjugate of X(j) for these (i,j): (291,673), (350,86), (659,190)
X(3226) = trilinear pole of line X(2)X(649)

### X(3227) = SS(a → bc) OF X(106)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc/(2bc - ab - ac)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Trilinears for X(3227) are obtained by applying the symbolic substitution (a,b,c) → (bc,ca,ab) to trilinear coordinates of X(106). As a symbolic substitution, this mapping takes circumconics to circumconics. In particular, it maps circumcircle onto the Steiner circumellipse, which passes through X(3227).

X(3227) lies on the Steiner circumellipse and these lines: 1,190   2,668   28,648   57,664   81,99   88,239   105,666   274,670   291,519   335,2087   671,2787   903,1022   957,1992

X(3227) = reflection of X(i) in X(j) for these (i,j): (2,1015), (668,2)
X(3227) = isogonal conjugate of X(3230)
X(3227) = isotomic conjugate of X(536)
X(3227) = X(1646)-cross conjugate of X(513)
X(3227) = Steiner-circumellipse-antipode of X(668)
X(3227) = projection from Steiner inellipse to Steiner circumellipse of X(1015)
X(3227) = the point of intersection, other than A, B, and C, of the Steiner circumellipse and hyperbola {{A,B,C,X(1),X(2)}}
X(3227) = antipode of X(2) in hyperbola {{A,B,C,X(1),X(2)}}
X(3227) = trilinear pole of line X(2)X(513)

### X(3228) = SS(a → bc) OF X(111)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc/(2b2c2 - a2b2 - a2c2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Trilinears for X(3228) are obtained by applying the symbolic substitution (a,b,c) → (bc,ca,ab) to trilinear coordinates of X(111). As a symbolic substitution, this mapping takes circumconics to circumconics. In particular, it maps circumcircle onto the Steiner circumellipse, which passes through X(3228).

X(3228) lies on the Steiner circumellipse and these lines: 6,99   25,648   37,668   42,190   111,385   263,1992   290,2395   671,804

X(3228) = reflection of X(i) in X(j) for these (i,j): (2,1084), (670,2)
X(3228) = isogonal conjugate of X(3231)
X(3228) = isotomic conjugate of X(538)
X(3228) = X(1645)-cross conjugate of X(512)
X(3228) = crossdifference of every pair of points on the line X(887)X(888)
X(3228) = Steiner-circumellipse-antipode of X(670)
X(3228) = projection from Steiner inellipse to Steiner circumellipse of X(1084)
X(3228) = the point of intersection, other than A, B, and C, of the Steiner circumellipse and hyperbola {{A,B,C,X(2),X(6)}}
X(3228) = antipode of X(2) in hyperbola {{A,B,C,X(2),X(6)}}
X(3228) = trilinear pole of line X(2)X(512)

### X(3229) = ISOGONAL CONJUGATE OF X(3225)

Trilinears    a(a2b4 + a2c4 - b2c4 - c 2b4) : :

X(3229) lies on these lines: 1,893   2,39   32,1613   141,706   187,237   232,420   511,694

X(3229) = isogonal conjugate of X(3225)
X(3229) = X(i)-Ceva conjugate of X(j) for these (i,j): (385,511), (699,6)
X(3229) = crosspoint of X(i) and X(j) for these (i,j): (2,694), (6,699)
X(3229) = crosssum of X(i) and X(j) for these (i,j): (2,698), (6,385), (192,2238), (523,2086)
X(3229) = crossdifference of every pair of points on the line X(2)X(669)
X(3229) = complement of X(3978)
X(3229) = crossdifference of PU(148)
X(3229) = intersection of line X(2)X(39)[X(194)] and line through X(2)-Ceva conjugate of X(194) and X(194)-Ceva conjugate of X(2)

### X(3230) = ISOGONAL CONJUGATE OF X(3227)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2bc - ab - ac)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3230) is the point of intersection of the line X(1)X(6) and the trilinear polar of X(6); c.f. X(1323). (Randy Hutson, February 10, 2016)

X(3230) lies on these lines: 1,6   31,2242   39,1201   41,2241   99,2106   101,1914   106,292   111,2703   172,595   187,237   239,1016   519,2238   672,1015   739,898   869,2177   992,2321   995,2276   1018,1575   1125,2295   1193,1500   1197,1621

X(3230) = reflection of X(3125) in X(3290)
X(3230) = isogonal conjugate of X(3227)
X(3230) = X(i)-Ceva conjugate of X(j) for these (i,j): (739,6), (898,667)
X(3230) = crosspoint of X(i) and X(j) for these (i,j): (6,739), (898,1016)
X(3230) = crosssum of X(i) and X(j) for these (i,j): (2,536), (513,1646), (891,1015)
X(3230) = crossdifference of every pair of points on the line X(2)X(513)
X(3230) = bicentric sum of PU(26)
X(3230) = PU(26)-harmonic conjugate of X(667)
X(3230) = inverse-in-Parry-isodynamic-circle of X(5040); see X(2)

### X(3231) = ISOGONAL CONJUGATE OF X(3228)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2b2c2 - a2b2 - a2c2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3231) lies on these lines: 2,6   23,2076   110,1691   111,694   187,237   468,2211

X(3231) = reflection of X(3124) in X(3291)
X(3231) = isogonal conjugate of X(3228)
X(3231) = X(729)-Ceva conjugate of X(6)
X(3231) = crosssum of X(i) and X(j) for these (i,j): (2,538), (512,1645), (888,1084)
X(3231) = crossdifference of every pair of points on the line X(2)X(512)
X(3231) = trilinear pole of line X(887)X(888)
X(3231) = inverse-in-Parry-isodynamic-circle of X(5027); see X(2)

### X(3232) = 2nd TRISECTED PERIMETER POINT

Trilinears       x : y : z (see below)
Barycentrics  ax : by : cz

There exist points A', B', C' on segments BC, CA, AB, respectively, such that B'C + C'B = C'A + A'C = A'B + B'A = (a + b + c)/3, and the lines AA', BB', CC' concur in X(3232). Near the beginning of the 21st century, trilinears x : y : z were found for X(3232) in terms of the unique real root of a cubic polynomial related to the cubic polynomial shown at X(369), the 1st trisected perimeter point. The proof is given in Sadi Abu-Saymeh, Mawaffaq Hajja, and Hellmuth Stachel, "Another cubic associated with the triangle," forthcoming in Journal for Geometry and Graphics.

If you have The Geometer's Sketchpad, you can view X(3232).

X(3232) = isotomic conjugate of X(369)

### X(3233) = VERTEX OF THE INSCRIBED KIEPERT PARABOLA

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(a2 - b2)(a2 - c2)[-2a4 + (b2 + c2)a2 + (b2 - c2)2]2
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Contributed by Fred Lang, April 1, 2007.

If you have The Geometer's Sketchpad, you can view Kiepert Inscribed Parabola.

X(3233) lies on these lines: 30,113   99,1304   110,476   114,468   355,3109   1302,1576

X(3233) = intercept of pedal and antipedal lines of X(110)

### X(3234) = VERTEX OF THE INSCRIBED YFF PARABOLA

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(2a3 - b3 - c3 - a2b - a2c + b2c + bc2)2/(b - c)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3234) lies on these lines: 101,514   118,516   1633,2736

Propetries of the inscribed Yff parabola: center = X(514); perspector = X(190), focus = X(101); directrix = X(4)X(9), the trilinear pole of X(1897). The parabola passes through these points: X(514), X(649), X(3234), X(3239). The axis of the parabola is X(101)X(514), and the Simson line of the focus is X(118)X(516). Contributed by Peter Moses, April 6, 2007.

If you have The Geometer's Sketchpad, you can view Yff Inscribed Parabola.

### X(3235) = INTERSECTION OF LINES X(1)X(32) and X(3)X(2007)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1 + sin(A - ω)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3235) lies on these lines: 1,32   3,2007   6,2008   55,1688   56,1687   1124,1342   1335,1343

### X(3236) = INTERSECTION OF LINES X(1)X(32) and X(3)X(2008)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1 - sin(A - ω)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3236) lies on these lines: 1,32   3,2008   6,2007   55,1687   56,1688   1124,1343   1335,1342

### X(3237) = INTERSECTION OF LINES X(1)X(256) and X(3)X(1673)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1 + cos(A + ω)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3237) lies on these lines: 1,256   3,1673   6,1672   55,1671   56,1670   371,3236   372,3235   1124,1689   1335,1690   1664,1675   1665,1674   2007,3103   2008,3102

### X(3238) = INTERSECTION OF LINES X(1)X(256) and X(3)X(1672)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1 - cos(A + ω)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3238) lies on these lines: 1,256   3,1672   6,1673   55,1670   56,1671   371,3235   372,3236   1124,1690   1335,1689   1664,1674   1665,1675   2007,3102   2008,3103

### X(3239) = ISOGONAL CONJUGATE OF X(1461)

Trilinears    (cos B - cos C)/(1 - cos A) : :
Trilinears    bc(b - c)(b + c - a)2 : :
Barycentrics    directed distance from A to Soddy line : :

X(3239) lies on the Yff parabola and these lines: 2,2400   9,652   100,3234   101,1309   111,2760   514,661

X(3239) = isogonal conjugate of X(1461)
X(3239) = isotomic conjugate of X(658)
X(3239) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,2968), (190,8), (341,2310), (346,1146), (644,2321), (2399,2804)
X(3239) = X(i)-cross conjugate of X(j) for these (i,j): (1146,346), (2310,341), (3119,200)
X(3239) = crosspoint of X(i) and X(j) for these (i,j): (2,1897), (8,190), (644,2321)
X(3239) = crosssum of X(i) and X(j) for these (i,j): (6,1459), (56,649), (513,2262)
X(3239) = crossdifference of every pair of points on the line X(31)X(56)
X(3239) = complement of X(4025)
X(3239) = perspector of circumconic centered at X(2968) (hyperbola {{A,B,C,X(8),X(75)}})
X(3239) = center of circumconic that is locus of trilinear poles of lines passing through X(2968)
X(3239) = intersection of trilinear polars of X(8) and X(75)

### X(3240) = INTERSECTION OF LINES X(1)X(2) AND X(6)X(100)

Trilinears    bc - 2a(b + c) : :

X(3240) lies on these lines: 1,2   6,100   44,751   45,2238   81,1376   88,1002   89,291   192,872   218,1005   238,2177   1013,1783

X(3240) = crossdifference of every pair of points on the line X(649)X(891)

### X(3241) = REFLECTION OF X(2) IN X(1)

Trilinears    bc(5a - b - c) : :
Trilinears    3 r - R sin B sin C : : : :
Barycentrics  5a - b - c : 5b - c - a : 5c - a - b
X(3241) = 2*X(1) - X(2) = 5 X(8) - 8 X(10)

X(3241) lies on these lines: 1,2   4,1392   6,644   7,528   30,944   149,1478   192,537   346,1449   376,517   381,952   390,527   518,1992   529,2098   956,1621   1000,2320   1222,2334

X(3241) = midpoint of X(2) and X(145)
X(3241) = reflection of X(i) in X(j) for these (i,j): (2,1), (8,2)
X(3241) = anticomplement of X(3679)
X(3241) = harmonic center of incircle and AC-incircle

### X(3242) = REFLECTION OF X(6) IN X(1)

Trilinears    2(a2 + b2 + c2) - a(a + b + c) : :
X(3242) = 2 X(1) - X(6)

X(3242) lies on these lines: 1,6   38,55   56,976   69,145   210,614   354,612   674,1469   982,1054   990,1350   1086,2550   1219,1257

X(3242) = midpoint of X(69) and X(145)
X(3242) = reflection of X(i) in X(j) for these (i,j): (6,1), (8,141), (69,145)
X(3242) = {X(1),X(9)}-harmonic conjugate of X(1279)

### X(3243) = REFLECTION OF X(9) IN X(1)

Trilinears    a2 + 3b2 + 3c2 - 2bc - 4ca - 4ab : :
X(3243) = 2 X(1) - X(9)

Let A'B'C' be the reflection of ABC in X(1). ABC and A'B'C' intersect at 6 points, which lie on an ellipse centered at X(1) with perspector X(10390). Let A" be the intersection of the tangents to this ellipse at the points where it intersects BC, and define B" and C" cyclically. (i.e., A"B"C" is the polar triangle of the ellipse.) A'B'C' and A"B"C" are perspective at X(3243). (Randy Hutson, August 29, 2018)

X(3243) lies on these lines: 1,6   7,145   57,100   65,2136   390,527   516,944   942,1706   971,1482   1056,2550

X(3243) = midpoint of X(7) and X(145)
X(3243) = reflection of X(i) in X(j) for these (i,j): (8,142), (9,1), (2136,3174)

### X(3244) = REFLECTION OF X(10) IN X(1)

Trilinears    bc(4a - b - c) : :
Trilinears    5 r - 2 R sin B sin C : :
Barycentrics  4a - b - c : 4b - c - a : 4c - a - b
X(3244) = 5 X(1) - 3 X(2) = 3 X(1) - X(8) = 2 X(8) - 3 X(10)

X(3244) lies on these lines: 1,2   6,2325   9,2137   58,643   65,1317   79,1320   382,515   516,944   517,550   546,946   950,2098   996,2334   1018,1475   1100,2321   1126,1222

X(3244) = midpoint of X(1) and X(145)
X(3244) = reflection of X(i) in X(j) for these (i,j): (8,1125), (10,1)
X(3244) = {X(1),X(2)}-harmonic conjugate of X(3636)
X(3244) = {X(1),X(8)}-harmonic conjugate of X(1125)

### X(3245) = REFLECTION OF X(1) IN ITS TRILINEAR POLAR

Trilinears    2b3 + 2c3 - a3 - 2(b + c)(a2 + bc) + a(b2 + c2 + 3bc) : :

X(3245) lies on these lines: 1,3   8,535   80,516

X(3245) = reflection of X(i) in X(j) for these (i,j): (1,1155), (36,484)
X(3245) = X(23)-of-reflection-triangle-of-X(1)
X(3245) = X(10295)-of-excentral-triangle
X(3245) = endo-homothetic center of Ehrmann vertex-triangle and X(2)-Ehrmann triangle; the homothetic center is X(7574)

### X(3246) = MIDPOINT OF X(1) AND X(44)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 4a2 + b2 + c2 - 4bc - ca - ab
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3246) lies on these lines: 1,6   88,105   513,1960   651,1319   678,899   752,1125

X(3246) = midpoint of X(i) and X(j) for these (i,j): (1,44), (238,1279)
X(3246) = crossdifference of every pair of points on the lnie X(45)X(513)

### X(3247) = INTERSECTION OF LINES X(1)X(6) AND X(2)X(2321)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a + 3b + 3c
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3247) lies on these lines: 1,6   2,2321   187,2959   226,347   281,1886   519,966   574,988   579,1334   594,1698   612,1962   902,968   1068,1826   1255,1796

X(3247) = crosssum of X(6) and X(3303)

### X(3248) = CROSSSUM OF X(1) AND X(190)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b - c)2
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3248) lies on these lines: 1,190   6,292   31,692   42,678   75,87   86,2665   239,2234   244,659   341,979   560,604   662,741   757,1178   798,1084   890,1977   922,2210   1015,1960   1100,2309

X(3248) = X(i)-Ceva conjugate of X(j) for these (i,j):
(1,649), (6,798), (31,667), (87,513), (604,1919), (649,3249), (873,1019), (979,650), (2297,2484), (2665,659)

X(3248) = X(i)-cross conjugate of X(j) for these (i,j): (3121,1015), (3249,649)
X(3248) = crosspoint of X(i) and X(j) for these (i,j): (1,649), (31,667), (873,1019), (1015,1357)
X(3248) = crosssum of X(i) and X(j) for these (i,j): (1,190), (75,668), (341,646), (872,1018)
X(3248) = crossdifference of every pair of points on the line X(190)X(646)

### X(3249) = SS(a → bc) OF X(764)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (ab - ac)3
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3249) lies on these lines: 1,649   32,1919   213,667   330,514

X(3249) = X(i)-Ceva conjugate of X(j) for these (i,j): (649,3248), (1919,1977)
X(3249) = crosspoint of X(i) and X(j) for these (i,j): (649,3248), (1919,1977)
X(3249) = crossdifference of every pair of points on the line X(350)X(899)

### X(3250) = INTERSECTION OF X(37)X(513) AND X(187)X(237)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b3 - c3)

X(3250) lies on these lines: 37,513   187,237   514,661   789,795   798,795

Let L be the line P(6)U(6) = X(37)X(513). Let M be the trilinear polar of the cevapoint of PU(6), this point being X(256). Let V = P(6)-Ceva conjugate of U(6) and W = U(6)-Ceva conjugate of P(6). The lines L, M, VW concur in X(3250). (Randy Hutson, December 26, 2015)

X(3250) = reflection of X(649) in X(665)
X(3250) = X(i)-Ceva conjugate of X(j) for these (i,j): (825,6), (2186,2170)
X(3250) = crosspoint of X(6) and X(825)
X(3250) = crosssum of X(2) and X(824)
X(3250) = crossdifference of every pair of points on the line X(2)X(31)
X(3250) = isogonal conjugate of X(4586)

### X(3251) = INTERSECTION OF X(1)X(513) AND X(42)X(663)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)(2a - b - c)2
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3251) lies on these lines: 1,513   42,663   55,667   678,1635   926,1642

X(3251) = reflection of X(1635) in X(1960)
X(3251) = isogonal conjugate of X(4618)
X(3251) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,2087), (100,44)
X(3251) = crosspoint of X(i) and X(j) for these (i,j): (1,1023), (44,100)
X(3251) = crosssum of X(i) and X(j) for these (i,j): (1,1022), (88,513), (100,3257)
X(3251) = crossdifference of every pair of points on the line X(44)X(88)

### X(3252) = INTERSECTION OF X(7)X(192) AND X(37)X(513)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 - ab - ac)/(a2 - bc)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3252) lies on these lines: 6,292   7,192   9,660   37,513   59,604   291,1002   813,840   984,2113

X(3252) = X(292)-Ceva conjugate of X(672)
X(3252) = crossdifference of every pair of points on the line X(238)X(812)

### X(3253) = INTERSECTION OF X(6)X(190) AND X(238)X(874)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(a2 - bc)/(b2c - b2a + c2b - c2a)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3253) lies on these lines: 2,2109   6,190   238,874   727,789   765,2209

### X(3254) = REFLECTION OF X(9) IN X(11)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c - a)/(bc - (b + c - a)2)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3254) lies on these lines: 1,528   4,2801   7,149   9,11   80,518   100,142   104,516   294,2323   320,2481   390,2320   527,1156   943,1125   1000,2550

X(3254) = midpoint of X(7) and X(149)
X(3254) = reflection of X(i) in X(j) for these (i,j): (9,11), (100,142)
X(3254) = isogonal conjugate of X(2078)
X(3254) = antigonal conjugate of X(9)

### X(3255) = INTERSECTION OF X(104)X(551) AND X(390)X(1392)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c - a)/(bc + (b + c - a)2)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3255) lies on these lines: 105,551   390,1392   527,2346   758,1000

X(3255) = isogonal conjugate of X(3256)

### X(3256) = ISOGONAL CONJUGATE OF X(3255)

Trilinears    a (b c + (b + c - a)^2)/(b + c - a) : :

X(3256) lies on these lines: 1,3   42,109   100,226   1174,1200   1259,1706

X(3256) = isogonal conjugate of X(3255)

### X(3257) = ISOGONAL CONJUGATE OF X(1635)

Trilinears    1/((b - c)(b + c - 2a)) : :

X(3257) lies on these lines: 44,88   100,513   106,238   320,908   527,666   658,1275   662,1019   758,1168   897,1757

Let P and Q be the intersections of line BC and circle {X(3),2R}. Let A' be the circumcenter of triangle PQX(3), and define B' and C' cyclically. The lines AA', BB', CC' concur in X(3257). (Randy Hutson, December 26, 2015)

Let A'B'C' and A"B"C" be the Ursa-minor and Ursa-major triangles, resp. Let A* be the reflection of A' in line B"C", and define B* and C* cyclically. Let A** be the reflection of A" in line B'C', and define B** and C** cyclically. Let A# = B*B**/\C*C**, and define B# and C# cyclically. The lines AA#, BB#, CC# concur in X(3257). (Randy Hutson, June 27, 2018)

X(3257) = reflection of X(i) in X(j) for these (i,j): (320,908), (3218,44)
X(3257) = isogonal conjugate of X(1635)
X(3257) = cevapoint of X(i) and X(j) for these (i,j): (1,1635), (88,1022), (100,1023), (649,1149)
X(3257) = X(i)-cross conjugate of X(j) for these (i,j): (44,765), (661,1168), (1022,88), (1023,100), (1635,1), (1769,75)
X(3257) = crossdifference of every pair of points on the line X(2087)X(3251)
X(3257) = isotomic conjugate of X(3762)
X(3257) = perspector of conic {A,B,C,PU(28)}
X(3257) = intersection of trilinear polars of P(28) and U(28)
X(3257) = trilinear pole of line X(1)X(88)

### X(3258) = CROSSSUM OF X(74) AND X(110)

Trilinears    (csc(A - B))/(cos B - 2 cos C cos A) + (csc(C - A))/(cos C - 2 cos A cos B)

Let P = X(477) and H=X(4); let HA be the orthocenter of the triangle BCP. Define HB and HC cyclically. Then H, HA, HB, HC are the vertices of a quadrilateral that is homothetic to the cyclic quadrilateral having vertices A, B, C, P. The center of homothety is X(3258). Moreover, X(3258) is the anticenter of the quadrilateral ABCP. (Randy Hutson, 9/23/2011)

Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Euler line. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A"B"C" be the reflection of A'B'C' in the Euler line of A'B'C' (line X(30)X(74)). The triangle A"B"C" is homothetic to ABC, with center of homothety X(3258); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)

X(3258) lies on the nine-point circle, the cevian circle of X(30), and these lines: 2,476   4,477   30,113   114,858   115,647   125,523   131,2072   132,468   133,403   232,1560   868,1649

X(3258) = midpoint of X(4) and X(477)
X(3258) = reflection of X(i) in X(j) for these (i,j); (125,3154), (1553,113)
X(3258) = complement of X(476)
X(3258) = complementary conjugate of X(526)
X(3258) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,1637), (4,526), (1138, 523)
X(3258) = crosspoint of X(2) and X(3268)
X(3258) = crosssum of X(74) and X(110)
X(3258) = crossdifference of every pair of points on the line X(2420)X(2433)
X(3258) = inverse-in-polar-circle of X(1304)
X(3258) = inverse-in-{circumcircle, nine-point circle}-inverter of X(842)
X(3258) = X(477)-of-Euler-triangle
X(3258) = reflection of X(125) in Euler line
X(3258) = perspector of circumconic centered at X(1637)
X(3258) = center of circumconic that is locus of trilinear poles of lines passing through X(1637)
X(3258) = intersection, other than X(126), of the nine-point circle of ABC and the Parry circle of the X(2)-Brocard triangle

### X(3259) = CROSSSUM OF X(100) AND X(104)

Barycentrics   (2*a - b - c)*(b - c)^2*(a^2*b - b^3 + a^2*c - 2*a*b*c + b^2*c + b*c^2 - c^3) : :
Trilinears   1/((a - b)(cos C + cos A - 1)) + 1/((c - a)(cos A + cos B - 1)) : :

X(3259) is the center of the rectangular hyperbola that passes through the points A, B, C, and X(56). (Randy Hutson, 9/23/2011)

X(3259) lies on the Sherman line (3259,3326); see http://forumgeom.fau.edu/FG2012volume12/FG201220.pdf)

A construction see Aris Pavlakis and Peter Moses, Hyacinthos 28741.

X(3259) lies on the nine-point circle, the Yff contact circle, and these lines: {2, 901}, {4, 953}, {11, 513}, {12, 13756}, {25, 10016}, {36, 855}, {56, 17101}, {114, 1281}, {115, 661}, {116, 3835}, {119, 517}, {120, 5087}, {121, 3814}, {133, 5146}, {149, 14513}, {153, 14511}, {226, 24201}, {244, 6615}, {1319, 1846}, {1566, 6544}, {1647, 5516}, {1878, 22835}, {2969, 20620}, {3120, 7336}, {3326, 10017}, {4370, 5513}, {4404, 24026}, {5954, 5993}, {5957, 11792}, {7951, 23153}, {8286, 8819}, {15614, 21252}, {17036, 20096}, {17605, 23152}, {24250, 25760}

X(3259) = midpoint of X(i) and X(j) for these {i,j}: {4, 953}, {56, 17101}, {149, 14513}, {153, 14511}
X(3259) = complement of X(901)
X(3259) = anticomplement of X(22102)
X(3259) = reflection of X(i) and X(j) for these {i,j}: {901, 22102}, {3937, 14115}, {6073, 119}, {6075, 11}
X(3259) = reflection of X(3937) in the line X(1)X(3)
X(3259) = {X(2),X(901)}-harmonic conjugate of X(22102)
X(3259) = polar circle inverse of X(1309)
X(3259) = orthoptic circle of the Steiner inellipe inverse of X(2726)
X(3259) = complement of the isogonal of X(900)
X(3259) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 900}, {2, 4928}, {6, 3960}, {31, 3310}, {42, 21894}, {44, 514}, {244, 1647}, {513, 519}, {514, 3834}, {519, 513}, {522, 5123}, {649, 16610}, {661, 3936}, {667, 8610}, {678, 6544}, {693, 21241}, {765, 6550}, {876, 25351}, {900, 10}, {902, 650}, {1019, 4395}, {1023, 4422}, {1319, 522}, {1404, 905}, {1635, 2}, {1639, 3452}, {1647, 11}, {1877, 521}, {1960, 37}, {2087, 1086}, {2161, 21198}, {2251, 6586}, {2325, 20317}, {2429, 25097}, {3251, 4370}, {3264, 21260}, {3285, 14838}, {3669, 17067}, {3689, 4521}, {3762, 141}, {3911, 4885}, {3943, 4129}, {4120, 1211}, {4358, 3835}, {4448, 17793}, {4530, 26932}, {4730, 1213}, {4768, 1329}, {4893, 27751}, {4895, 9}, {5440, 20315}, {6544, 16594}, {14407, 16589}, {14408, 6376}, {14427, 6554}, {14429, 21530}, {14437, 13466}, {14584, 3738}, {16704, 4369}, {17780, 24003}, {21805, 661}, {22086, 1214}, {23344, 24036}, {23703, 3035}, {23757, 119}, {23838, 3036}, {24004, 27076}
X(3259) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 3310}, {4, 900}, {11, 1647}, {513, 6550}, {1145, 23757}
X(3259) = crosssum of X(100) and X(104)
X(3259) = crossdifference of every pair of points on the line X(2423)X(2427)
X(3259) = crosspoint of X(i) and X(j) for these (i,j): {513, 517}, {900, 14584}, {1145, 23757}
X(3259) = X(i)-isoconjugate of X(j) for these (i,j): {104, 9268}, {765, 10428}, {909, 5376}
X(3259) = inverse-in-polar-circle of X(1309)
X(3259) = perspector of circumconic centered at X(3310)
X(3259) = center of circumconic that is locus of trilinear poles of lines passing through X(3310); this conic is a rectangular circumhyperbola that is isogonal conjugate of line X(3)X(8)
X(3259) = barycentric product X(i) X(j) for these {i,j}: {514, 23757}, {900, 10015}, {908, 1647}, {1086, 1145}, {1769, 3762}, {1846, 26932}, {2087, 3262}, {2397, 6550}, {4120, 23788}, {4530, 22464}, {17205, 21942}
X(3259) = barycentric quotient X(i) / X(j) for these {i,j}: {517, 5376}, {900, 13136}, {1015, 10428}, {1145, 1016}, {1769, 3257}, {2087, 104}, {2183, 9268}, {2397, 6635}, {2427, 6551}, {2804, 4582}, {3310, 901}, {6550, 2401}, {8661, 2423}, {10015, 4555}, {23757, 190}, {23788, 4615}

### X(3260) = ISOTOMIC CONJUGATE OF X(74)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b2c2(cos A - 2 cos B cos C)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3260) lies on these lines: 4,69   94,323   98,2855   99,477   183,1995   290,892   298,300   298,301   324,394   325,523   328,1494   338,524   1138,1272

X(3260) = isotomic conjugate of X(74)
X(3260) = anticomplement of X(3003)
X(3260) = X(328)-Ceva conjugate of X(311)
X(3260) = cevapoint of X(i) and X(j) for these (i,j): (2,146), (69,1272), (323,2071)
X(3260) = X(113)-cross conjugate of X(2)
X(3260) = crosspoint of X(300) and X(301)
X(3260) = crossdifference of every pair of points on the line X(32)X(3049)

### X(3261) = ISOTOMIC CONJUGATE OF X(101)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)/a3
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3261) lies on these lines: 2,2412   75,522   86,1459   99,2690   100,2860   325,523   514,1921   798,802   824,1577

X(3261) = isotomic conjugate of X(101)
X(3261) = X(i)-Ceva conjugate of X(j) for these (i,j): (561,1111), (668,1233), (670,1269), (1978,76)
X(3261) = cevapoint of X(i) and X(j) for these (i,j): (2,150), (850,1577)
X(3261) = X(i)-cross conjugate of X(j) for these (i,j): (116,2), (1111,561), (1577,693), (1978,76)
X(3261) = crosspoint of X(76) and X(1978)
X(3261) = crosssum of X(i) and X(j) for these (i,j): (32,1919), (213,3063), (1980,2205)
X(3261) = crossdifference of every pair of points on the line X(32)X(560)

### X(3262) = ISOTOMIC CONJUGATE OF X(104)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc2A)(-1 + cos B + cos C)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3262) lies on these lines: 7,8   92,345   99,2687   100,2861   105,2865   264,1969   286,1792   311,313   314,1389   325,523   1111,1266   1232,1269   1332,1944

X(3262) = isotomic conjugate of X(104)
X(3262) = cevapoint of X(2) and X(153)
X(3262) = X(119)-cross conjugate of X(2)
X(3262) = crosssum of X(2175) and X(2251)
X(3262) = crossdifference of every pair of points on the line X(32)X(3063)

### X(3263) = ISOTOMIC CONJUGATE OF X(105)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 + c2 - ab - ac)/a2
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3263) lies on these lines: 2,37   8,304   10,1930   85,341   99,1325   100,2862   104,2865   109,2866   274,1390   305,561   313,1233   314,2346   325,523   442,1228   742,2238

X(3263) = isotomic conjugate of X(105)
X(3263) = anticomplement of X(3290)
X(3263) = X(120)-cross conjugate of X(2)
X(3263) = crosspoint of X(75) and X(334)
X(3263) = crosssum of X(31) and X(2210)
X(3263) = crossdifference of every pair of points on the line X(32)X(667)

### X(3264) = ISOTOMIC CONJUGATE OF X(106)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (2a - b - c)/a3
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3264) lies on these lines: 10,75   86,1222   99,2758   100,2863   190,2183   320,668   325,523   350,899   730,2234   1240,1268

X(3264) = isotomic conjugate of X(106)
X(3264) = X(121)-cross conjugate of X(2)
X(3264) = crossdifference of every pair of points on the line X(32)X(1919)

### X(3265) = ISOTOMIC CONJUGATE OF X(107)

Trilinears    (csc2A)(cos A)(sin 2B - sin 2C)
Barycentrics    cot A (tan B - tan C) : :
Barycentrics    (b^2 - c^2) (b^2 + c^2 - a^2)^2 : :
Barycentrics    directed distance from A to van Aubel line : :

X(3265) lies on the Kiepert Parabola and these lines: 2,2419   99,1304   325,523   441,525   2501,2799

X(3265) = midpoint of X(647) and X(2525)
X(3265) = isotomic conjugate of X(107)
X(3265) = X(i)-Ceva conjugate of X(j) for these (i,j): (99,69), (3267,525)
X(3265) = X(i)-cross conjugate of X(j) for these (i,j): (122,2), (520,525), (2972,394)
X(3265) = crosspoint of X(69) and X(99)
X(3265) = crosssum of X(i) and X(j) for these (i,j): (6,2485), (25,512), (1974,2489)
X(3265) = crossdifference of every pair of points on the line X(25)X(32)
X(3265) = anticomplement of X(6587)
X(3265) = perspector of hyperbola {{A,B,C,X(69),X(76)}} (the isotomic conjugate of the van Aubel line)
X(3265) = pole wrt polar circle of trilinear polar of X(6529) (line X(25)X(393))
X(3265) = polar conjugate of X(6529)

### X(3266) = ISOTOMIC CONJUGATE OF X(111)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (2a2 - b2 - c2)/a3
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3266) lies on these lines: 2,39   23,99   98,2858   110,2868   325,523   1975,1995

X(3266) = isotomic conjugate of X(111)
X(3266) = anticomplement of X(3291)

### X(3267) = ISOTOMIC CONJUGATE OF X(112)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 - c2)(b2 + c2 - a2)/a3
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3267) is the trilinear pole of the line X(125)X(339.   Randy Hutson, August 15, 2013

X(3267) lies on these lines: 2,2485   99,935   100,2859   325,523

X(3267) = isotomic conjugate of X(112)
X(3267) = anticomplement of X(2485)
X(3267) = X(i)-Ceva conjugate of X(j) for these (i,j): (305,339), (670,1502)
X(3267) = X(i)-cross conjugate of X(j) for these (i,j): (127,2), (339,305()
X(3267) = crosspoint of X(i) and X(j) for these (i,j): (99,1799), (670,1502)
X(3267) = crosssum of X(i) and X(j) for these (i,j): (32,3049), (512,1843), (669,1501), (2489,3199)
X(3267) = crossdifference of every pair of points on the line X(32)X(682)

### X(3268) = ISOTOMIC CONJUGATE OF X(476)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc2A)(sin(B - C))(1 + 2 cos 2A)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3268) lies on these lines: 2,1637   99,110   147,2793   325,523   339,2972   525,1636   1297,2373   2394,2986

X(3268) = isogonal conjugate of X(14560)
X(3268) = isotomic conjugate of X(476)
X(3268) = anticomplement of X(1637)
X(3268) = X(3258)-cross conjugate of X(2)
X(3268) = crosspoint of X(99) and X(1494)
X(3268) = crosssum of X(512) and X(1495)
X(3268) = crossdifference of every pair of points on the line X(32)X(3124)

### X(3269) = CROSSPOINT OF X(6) AND X(647)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 - c2)2(b2 + c2 - a2)2
Trilinears    sin A cos^2 A sin^2(B - C) : :
Trilinears    cos A sin 2A sin^2(B - C) : :

X(3269) lies on the Brocard inellipse, the inconic with perspector X(2052), and on these lines: 3,248   4,1987   6,74   32,1204   39,185   64,2207   99,287   115,125   184,574   186,1971   339,525   1425,1500   1636,2972   1899,2549

X(3269) = reflection of X(3331) in X(232)
X(3269) = X(i)-Ceva conjugate of X(j) for these (i,j): (6,647), (64,512), (66,3005), (287,684), (394,520), (1177,351), (2052, 523)
X(3269) = crosspoint of X(6) and X(647)
X(3269) = crosssum of X(i) and X(j) for these (i,j): (2,648), (3,1625), (4,112), (6,1624), (99,315), (107,393), (110,577), (162,2326)
X(3269) = crossdifference of every pair of points on the line X(107)X(110)
X(3269) = X(92)-isoconjugate of X(250)
X(3269) = crosssum of X(2479) and X(2480)
X(3269) = barycentric square of X(656)

### X(3270) = X(1)-CEVA CONJUGATE OF X(647)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b - c)2(b + c - a)2(b2 + c2 - a2)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3270) lies on the orthic inconic and on these lines: 1,185   3,1813   11,125   25,2192   33,51   55,184   56,1204   64,1398   103,1461   217,1500   497,1899   511,3100   651,2808   926,2170   1069,1092   1409,2293   1827,2262   2637,2638   2876,3056

X(3270) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,647), (3,652), (4,650), (6,657), (64,649), (74,654), (2192,663)
X(3270) = crosspoint of X(i) and X(j) for these (i,j): (1,1021), (3,652), (4,650), (6,1459), (1146,2968)
X(3270) = crosssum of X(i) and X(j) for these (i,j): (1,1020), (2,1897), (3,651), (4,653), (7,934), (100,329), (108,278)
X(3270) = crossdifference of every pair of points of the line X(651)X(653)
X(3270) = orthic-isogonal conjugate of X(650)
X(3270) = perspector of orthic triangle and tangential triangle of Feuerbach hyperbola
X(3270) = X(92)-isoconjugate of X(1262)

### X(3271) = CROSSSUM OF X(2) AND X(100)

Trilinears    a(b - c)2(b + c - a) : :
Trilinears    a^2 (1 - cos(B - C)) : :
Trilinears    a^2 sin^2(B/2 - C/2) : :
Trilinears    a^2 csc^2(B/2 - C/2) : :

X(3271) lies on the Mandart inellipse and on these lines: 1,2810   6,692   8,646   9,3056   11,124   21,1682   25,1397   31,51   44,674   55,2316   56,1461   65,2835   100,3030   105,651   215,2194   238,511   244,1357   373,750   389,3073   512,2643   513,1086   614,1401   926,2170   1015,1960   1083,1332   1361,1411   1365,2969   1428,3220   1843,2212   1977,3124   2092,2309   2161,2875   2183,2223   2293,2347

X(3271) = isogonal conjugate of X(4998)
X(3271) = X(i)-Ceva conjugate of X(j) for these (i,j): (6,3063), (8,650), (25,667), (31,512), (42,2488), (55,663), (56,649), (105,665), (244,1015), (263,788), (1411,3310), (2218,647)
X(3271) = crosspoint of X(i) and X(j) for these (i,j): (6,513), (55,663), (56,651), (244,2170)
X(3271) = crosssum of X(i) and X(j) for these (i,j): (2,100), (7,664), (8,190), (56,651), (480,644), (1259,1332)
X(3271) = crossdifference of every pair of points of the line X(190)X(644)
X(3271) = isogonal conjugate of X(4998)
X(3271) = X(59)-isoconjugate of X(75)
X(3271) = trilinear square of X(6729)

Points Associated with Equilateral Triangles

The points X(3272) to X(3283) are associated with six equilateral triangles related to the (1st) Morley triangle; trilinears for vertices of these triangles are given just below. Received from Milorad R. Stevanovic, Nov. 24, 2007 and Dec. 25, 2007.

1st Morley triangle, M1M2M3, where
M1 = 1 : 2 cos C/3 : 2 cos B/3
M2 = 2 cos C/3 : 1 : 2 cos A/3
M3 = 2 cos B/3 : 2 cos A/3 : 1

2nd Morley triangle, P1P2P3, where
P1 = 1 : 2 cos(C/3 - 2π/3) : 2 cos(B/3 - 2π/3)
P2 = 2 cos(C/3 - 2π/3) : 1 : 2 cos(A/3 - 2π/3)
P3 = 2 cos(B/3 - 2π/3) : 2 cos(A/3 - 2π/3) : 1

3rd Morley triangle, S1S2S3, where
S1 = 1 : 2 cos(C/3 - 4π/3) : 2 cos(B/3 - 4π/3)
S2 = 2 cos(C/3 - 4π/3) : 1 : 2 cos(A/3 - 4π/3)
S3 = 2 cos(B/3 - 4π/3) : 2 cos(A/3 - 4π/3) : 1

The triangles M1M2M3, P1P2P3, S1S2S3 are discussed in TCCT, page 165-166. The three are pairwise homothetic. See also
1st Morley triangle at MathWorld
2nd Morley triangle at MathWorld
3rd Morley triangle at MathWorld

Unique equilateral triangle inscribed in ABC and homothetic to the 1st Morley triangle, J1J2J3, where
J1 = 0 : sin(A/3 - C/3 + π/3) : sin(A/3 - B/3 + π/3)
J2 = sin(B/3 - C/3 + π/3) : 0 : sin(B/3 - A/3 + π/3)
J3 = sin(C/3 - B/3 + π/3) : sin(C/3 - A/3 + π/3) : 0

Circumtangential triangle, inscribed in the circumcircle of ABC and homothetic to the 1st Morley triangle, T1T2T3. For trilinears, of the vertices, see TCCT, page 166, or MathWorld. Trilinears found by M. Stevanovic:

T1 = sin(B/3 - A/3 + π/3) sin(C/3 - A/3 + π/3) : sin(C/3 - B/3) sin(B/3 - A/3 + π/3) : sin(B/3 - C/3) sin(C/3 - A/3 + π/3)
T2 = sin(C/3 - A/3) sin(A/3 - B/3 + π/3) : sin(C/3 - B/3 + π/3) sin(A/3 - B/3 + π/3) : sin(A/3 - C/3) sin(C/3 - B/3 + π/3)
T3 = sin(B/3 - A/3) sin(A/3 - C/3 + π/3) : sin(A/3 - B/3) sin(B/3 - C/3 + π/3) : sin(A/3 - C/3 + π/3) sin(B/3 - C/3 + π/3)

Circumnormal triangle inscribed in the circumcircle of ABC and homothetic to the 1st Morley triangle, N1N2N3. For trilinears, of the vertices, see TCCT, page 166, or MathWorld. Trilinears found by M. Stevanovic:

N1 = - cos(B/3 - A/3 + π/3) cos(C/3 - A/3 + π/3) : cos(B/3 - C/3) cos(B/3 - A/3 + π/3) : cos(B/3 - C/3) cos(C/3 - A/3 + π/3)
N2 = cos(C/3 - A/3) cos(A/3 - B/3 + π/3) : - cos(C/3 - B/3 + π/3) cos(A/3 - B/3 + π/3) : cos(A/3 - A/3) cos(C/3 - B/3 + π/3)
N3 = cos(A/3 - B/3) cos(A/3 - C/3 + π/3) : cos(A/3 - B/3) cos(B/3 - C/3 + π/3) : - cos(A/3 - C/3 + π/3) cos(B/3 - C/3 + π/3)

Peter Moses notes (April, 14, 2008) that, regarding the reference triangle R and the 6 equilateral triangles (here abbreviated as M, P, S, T, N, J), there are 21 pairs, and each pair except {R,T}, {R,N}, {R, J} are perspective. Perspectors for the remaining 18 pairs are given by this table:

Pair Perspector
R, M X(357)
R, P X(1136)
R, S X(1134)
M, P X(358)
M, S X(1135)
M, T X(3278)
M, N X(3279)
M, J X(3273)
P, S X(1137)
P, T X(3280)
P, N X(3281)
P, J X(3274)
S, T X(3282)
S, N X(3283)
S, J X(3275)
T, N X(3)
T, J X(3334)
N, J X(3335)
Centers of the six equilateral triangles are as follows:
Triangle Center
M X(356)
P X(3276)
S X(3277)
T X(3)
N X(3)
J X(3272)

### X(3272) = CENTER OF EQUILATERAL TRIANGLE J1J2J3

Trilinears    cos(B/3 - C/3) : :

Let A1B1C1, A2B2C2, A3B3C3 be the 1st Morley, 2nd Morley and 3rd Morley triangles of ABC, respectively. Then {A, A1, A2, A3} are concyclic on a circle, A0.
{B, B1, B2, B3} are concyclic on a circle, B0.
{C, C1, C2, C3} are concyclic on a circle C0.
The radical center of A0, B0, C0 is X(3272). (César Lozada, June 16, 2018)

X(3272) lies on these lines: 356,3273   357,1135   358,1136   396,523   1134,1137   3274,3276   3275,3277

### X(3273) = PERSPECTOR OF TRIANGLES M1M2M3 AND J1J2J3

Trilinears    sin(A/3 + π/3) : :

Let A'B'C' be the 1st Morley triangle. Let La be the line through A' parallel to BC, and define Lb and Lc cyclically. Let A" = Lb∩Lc, B" = Lc∩La, C" = La∩Lb. The lines AA", BB", CC" concur in X(3273). (Randy Hutson, July 20, 2016)

X(3273) lies on these lines: 16,358   356,3272   357,1136

X(3273) = crosssum of X(1135) and X(3274)
X(3273) = isogonal conjugate of X(3604)
X(3273) = perspector of ABC and unary cofactor triangle of 3rd Morley triangle

### X(3274) = PERSPECTOR OF TRIANGLES P1P2P3 AND J1J2J3

Trilinears    cos(A/3 + π/6) : :
Trilinears    sin(A/3 - π/3) : :

Let A'B'C' be the 1st Morley triangle. Let La be the trilinear polar of A', and define Lb, Lc cyclically. Let A" = Lb∩Lc, B" = Lc∩La, C" = La∩Lb. The lines AA", BB", CC" concur in X(3274). (Randy Hutson, July 20, 2016)

X(3274) lies on these lines: 16,358   1134,1136   3272,3276

X(3274) = isogonal conjugate of X(3602)
X(3274) = crosssum of X(358) and X(3275)
X(3274) = crossdifference of every pair of points on the perspectrix of ABC and 1st Morley triangle
X(3274) = perspector of ABC and unary cofactor triangle of 1st Morley triangle
X(3274) = perspector of ABC and unary cofactor triangle of 1st Morley adjunct triangle

### X(3275) = PERSPECTOR OF TRIANGLES S1S2S3 AND J1J2J3

Trilinears    sin(A/3) : :

X(3275) lies on these lines: 16,358   357,1134   3272,3277

X(3275) = isogonal conjugate of X(3603)
X(3275) = perspector of ABC and unary cofactor triangle of 2nd Morley triangle
X(3275) = perspector of ABC and unary cofactor triangle of 2nd Morley adjunct triangle
X(3275) = crosssum of X(1137) and X(3273)

### X(3276) = CENTER OF 2nd MORLEY TRIANGLE

Trilinears    cos(A/3 - 2π/3) + 2 cos(B/3 - 2π/3) cos(C/3 - 2π/3) : :
Trilinears    cos(A/3) - cos(B/3) cos(C/3) : :
Trilinears    sec(A/3) - sec(B/3) sec(C/3) : :
X(3276) = X(356) + X(3277) - X(3)

X(3276) lies on these lines: 3,3280   356,357   1136,1137   3272,3274

X(3276) = isogonal conjugate of X(3606)

### X(3277) = CENTER OF 3rd MORLEY TRIANGLE

Trilinears    cos(A/3 - 4π/3) + 2 cos(B/3 - 4π/3)cos(C/3 - 4π/3)

X(3277) = X(356) + X(3276) - X(3)

X(3277) lies on these lines: 3,3282   356,1134   1136,1137   3272,3275

X(3277) = isogonal conjugate of X(3607)

### X(3278) = PERSPECTOR OF TRIANGLES M1M2M3 AND T1T2T3

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(A/3) cos(2A/3) + cos(B/3) cos(C/3)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3278) lies on these lines: 3,356   16,358

### X(3279) = PERSPECTOR OF TRIANGLES M1M2M3 AND N1N2N3

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(B/3) cos(C/3) + sin(A/3) sin(2A/3)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3279) lies on these lines: 3,356   358,3281   1135,3283

### X(3280) = PERSPECTOR OF TRIANGLES P1P2P3 AND T1T2T3

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 2 cos(A/3) - cos A - 2 cos(B/3) cos(C/3)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3280) lies on these lines: 3,3276   16,358

### X(3281) = PERSPECTOR OF TRIANGLES P1P2P3 AND N1N2N3

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 2 cos(A/3) + cos A - 2 cos(B/3) cos(C/3)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3281) lies on these lines: 3,3276   358,3279   1137,3283

### X(3282) = PERSPECTOR OF TRIANGLES S1S2S3 AND T1T2T3

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos A + 2 sin(B/3) sin(C/3) - sin(A/3 - π/6)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3282) lies on these lines: 3,3277   16,358

### X(3283) = PERSPECTOR OF TRIANGLES S1S2S3 AND N1N2N3

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos A - 2 sin(B/3) sin(C/3) + sin(A/3 - π/6)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3283) lies on these lines: 3,3277   1135,3279   1137,3281

### X(3284) = INTERSECTION OF LINES X(2)X(340) AND X(3)X(6)

Trilinears    a(b2 + c2 - a2)[(b2 - c2)2 + a2(b2 + c2 - 2a2)] : :
Trilinears    (sin 2A) (cos A - 2 cos B cos C) : :
Trilinears    (sin 2A) (3 cos A - 2 sin B sin C) : :

Let L denote the line through X(4) perpendicular to the Euler line. Coefficients for a trilinear equation for L are the trilinears for X(3284).

X(3284) lies on these lines: 2,340   3,6   23,232   30,1990   112,2693   231,2072   248,895   393,3146   401,648   441,425   520,647

X(3284) = midpoint of X(401) and X(648)
X(3284) = complement of X(340)
X(3284) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,1511), (2420,1636)
X(3284) = X(1636)-cross conjugate of X(2420)
X(3284) = crosssum of X(i) and X(j) for these (i,j): (4,1990), (6,186), (125,1637), (281,860), (470,471)
X(3284) = crosspoint of X(2) and X(265)
X(3284) = crossdifference of every pair of points on the line X(4)X(523)
X(3284) = X(92)-isoconjugate of X(74)
X(3284) = perspector of circumconic centered at X(1511)
X(3284) = center of circumconic that is locus of trilinear poles of lines passing through X(1511)
X(3284) = inverse-in-MacBeath-circumconic of X(3)
X(3284) = {X(61),X(62)}-harmonic conjugate of X(389)

### X(3285) = INTERSECTION OF LINES X(3)X(6) AND X(21)X(45)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2a - b - c)/(b + c)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let L denote the line through X(10) perpendicular to the Euler line. Coefficients for a trilinear equation for L are the trilinears for X(3285).

X(3285) lies on these lines: 3,6   21,45   44,2251   55,2206   81,89   110,2384   112,953   649,834

X(3285) = isogonal conjugate of X(4080)
X(3285) = cevapoint of X(902) and X(2251)
X(3285) = crosspoint of X(81) and X(759)
X(3285) = crosssum of X(37) and X(758)
X(3285) = crossdifference of every pair of points on the line X(10)X(523)

### X(3286) = INTERSECTION OF LINES X(3)X(6) AND X(7)X(21)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 - ab - ac)/(b + c)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let L denote the line through X(37) perpendicular to the Euler line. Coefficients for a trilinear equation for L are the trilinears for X(3286).

X(3286) lies on these lines: 3,6   7,21   28,277   31,2274   36,238   55,81   63,2352   110,840   141,1009   198,1778   241,1876   283,1037   333,1376   518,2223   672,1818   741,813   759,1308   940,1011   958,1010

X(3286) = inverse-in-circumcircle of X(3110)
X(3286) = cevapoint of X(672) and X(2223)
X(3286) = X(2254)-cross conjugate of X(2283)
X(3286) = crosspoint of X(58) and X(741)
X(3286) = crosssum of X(10) and X(740)
X(3286) = crossdifference of every pair of points on the line X(37)X(523)

### X(3287) = INTERSECTION OF LINES X(44)X(513) AND X(523)X(879)

Trilinears    (b - c)(b + c - a)(a2 + bc) : :

Let L denote the line through X(1) parallel to the Brocard axis, X(3)X(6). Coefficients for a trilinear equation for L are the trilinears for X(3287).

X(3287) lies on these lines: 6,523   37,2605   44,513

X(3287) = X(i)-Ceva conjugate of X(j) for these (i,j): (83,11), (99,55), (666,385)
X(3287) = crosspoint of X(i) and X(j) for these (i,j): (9,645), (651,2298)
X(3287) = crosssum of X(1) and X(3287)
X(3287) = crossdifference of every pair of points on the line X(1)X(256)
X(3287) = bicentric difference of PU(88)
X(3287) = PU(88)-harmonic conjugate of X(1284)

### X(3288) = INTERSECTION OF LINES X(237)X(351) AND X(523)X(879)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 - c2)(a4 - 2b2c2 - a2b2 - a2c2)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let L denote the line through X(2) parallel to the Brocard axis, X(3)X(6). Coefficients for a trilinear equation for L are the trilinears for X(3288).

X(3288) lies on these lines: 6,523   111,2698   187,237   323,401   352,1499   419,2501

X(3288) = reflection of X(i) in X(j) for these (i,j): (2451,3049), (3049,3050)
X(3288) = crosspoint of X(99) and X(3114)
X(3288) = crosssum of X(512) and X(3117)
X(3288) = crossdifference of every pair of points on the line X(2)X(51)
X(3288) = radical center of {circumcircle, Brocard circle, orthosymmedial circle}

### X(3289) = INTERSECTION OF LINES X(2)X(6) AND X(3)X(217)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(b2 + c2 - a2)(b4 + c4 - a2b2 - a2c2)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let L denote the line through X(4) perpendicular to the Brocard axis, X(3)X(6). Coefficients for a trilinear equation for L are the trilinears for X(3289).

X(3289) lies on these lines: 2,6   3,217   30,1625   110,1971   112,2706   115,1568   184,418   232,511   520,647   571,1501

X(3289) = reflection of X(3331) in X(1625)
X(3289) = X(i)-Ceva conjugate of X(j) for these (i,j): (287,3), (2421,684)
X(3289) = crosspoint of X(3) and X(287)
X(3289) = crosssum of X(4) and X(232)
X(3289) = crossdifference of every pair of points on the line X(4)X(512)

### X(3290) = INTERSECTION OF LINES X(2)X(37) AND X(230)X(231)

Trilinears  b3 + c3 - 2abc + (a2 - bc)(b + c) : :

Let L denote the line through X(3) perpendicular to the line X(1)X(6). Coefficients for a trilinear equation for L are the trilinears for X(3290).

X(3290) lies on these lines: 1,2271   2,37   6,354   9,982   19,1611   25,1841   65,2176   105,910   111,1290   120,1738   172,1104   213,942   230,231   241,292   244,672   518,2238   1018,1739   1149,2170   1212,2275

X(3290) = midpoint of X(3125) and X(3230)
X(3290) = isogonal conjugate of X(2991)
X(3290) = complement of X(3263)
X(3290) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,120), (666,513)
X(3290) = crosspoint of X(2) and X(105)
X(3290) = crosssum of X(6) and X(518)
X(3290) = crossdifference of every pair of points on the line X(3)X(667)
X(3290) = perspector of circumconic centered at X(120)
X(3290) = center of circumconic that is locus of trilinear poles of lines passing through X(120)
X(3290) = PU(4)-harmonic conjugate of X(6591)
X(3290) = crossdifference of PU(44)

### X(3291) = INTERSECTION OF LINES X(2)X(39) AND X(230)X(231)

Trilinears    a(b4 + c4 + a2b2 + a2c2 - 4b2c2) : :
Trilinears    (sin 2A)(sin^2 A) cos(A + ω) : :
Trilinears    (cos A)(sin^3 A) cos(A + ω) : :

Let L denote the line through X(3) perpendicular to the line X(2)X(6). Coefficients for a trilinear equation for L are the trilinears for X(3291).

X(3291) lies on these lines: 2,39   6,373   23,111   25,1611   32,1995   51,1613   110,1692   115,858   230,231   325,1570

X(3291) = midpoint of X(3124) and X(3231)
X(3291) = complement of X(3266)
X(3291) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,126), (892,512)
X(3291) = crosssum of X(i) and X(j) for these (i,j): (6,524), (525,1648)
X(3291) = crossdifference of every pair of points on the line X(3)X(669)
X(3291) = X(92)-isoconjugate of X(290)
X(3291) = perspector of circumconic centered at X(126)
X(3291) = center of circumconic that is locus of trilinear poles of lines passing through X(126)
X(3291) = intersection of tangents to hyperbola {A,B,C,X(2),X(6)} at X(2) and X(111)
X(3291) = crosspoint of X(2) and X(111)
X(3291) = PU(4)-harmonic conjugate of X(2489)

### X(3292) = INTERSECTION OF LINES X(3)X(49) AND X(23)X(110)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 - a2)(b2 + c2 - 2a2)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let L denote the line through X(4) perpendicular to the line X(2)X(6). Coefficients for a trilinear equation for L are the trilinears for X(3292).

Let O* be the circle with segment X(15)X(16) as diameter (and center X(187). Let P be the perspector of O*. Then X(3292) is the trilinear pole of the polar of P with respect to O*. See X(5642) for a similar property involving the segment X(13)X(14). (Randy Hutson, July 18, 2014)

X(3292) lies on these lines: 2,575   3,49   6,237   23,110   51,576   112,2763   352,2030   450,648   520,647   539,2072   542,858   1209,1493

X(3292) = midpoint of X(110) and X(323)
X(3292) = reflection of X(1495) in X(1)
X(3292) = X(i)-Ceva conjugate of X(j) for these (i,j): (524,187), (895,3)
X(3292) = crosspoint of X(3) and X(895)
X(3292) = crosssum of X(4) and X(468)
X(3292) = crossdifference of every pair of points on the line X(4)X(1499)

### X(3293) = SS(a → b + c) OF X(3)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)[(c + a)2 + (a + b)2 - (b + c)2]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3293) lies on these lines: 1,2   37,762   40,209   55,1724   58,100   71,380   81,1126   191,1045   213,1018   484,1046   740,872   942,1739

X(3293) = reflection of X(1) in X(1193)
X(3293) = X(i)-Ceva conjugate of X(j) for these (i,j): (81,37), (83,3294), (1126,1)

### X(3294) = SS(a → ab + ac) OF X(3)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (ab + ac)[(bc + ba)2 + (ca + cb)2 - (ab + ac)2]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3294) lies on these lines: 1,6   21,101   190,274   573,962   672,1125   846,2664

X(3294) = X(i)-Ceva conjugate of X(j) for these (i,j): (83,3293), (86,42)

### X(3295) = INTERSECTION OF LINES X(1)X(3) AND X(4)X(390)

Trilinears        2 + cos A : 2 + cos B : 2 + cos C
Barycentrics   (2 + cos A)(sin A) : (2 + cos B)(sin B) : (2 + cos C)(sin C)

X(3295) lies on these lines: 1,3   4,390   5,497   11,498   12,381   20,1056   21,145   30,388   33,1598   34,1597   73,1480   77,1059   149,2476   172,1384   212,1497   221,500   284,2256   355,950   378,1398   382,1478   474,1387   496,1058   595,1126   601,1496   602,1263   902,1468   916,2293   943,1260   944,1012   1036,1807

X(3295) = midpoint of X(1) and X(1697)
X(3295) = isogonal conjugate of X(3296)
X(3295) = crosssum of X(1) and X(3338)
X(3295) = crossdifference of every pair of points of the line X(650)X(2423)
X(3295) = extangents-to-intangents similarity image of X(3)
X(3295) = center of circle that is locus of crosssums of incircle antipodes
X(3295) = X(1595)-of-excentral-triangle
X(3295) = X(1593)-of-2nd-circumperp-triangle
X(3295) = X(11414)-of-1st-circumperp-triangle
X(3295) = excentral-to-2nd-circumperp similarity image of X(1697)
X(3295) = homothetic center of inner Yff triangle and cross-triangle of ABC and 2nd Johnson-Yff triangle
X(3295) = {X(i),X(j)}-harmonic conjugate of X(k) for thewe (i,j,k): (1,3,999), (1,40,942), (55,56,35)

### X(3296) = ISOGONAL CONJUGATE OF X(3295)

Trilinears    1/(2 + cos A) : 1/(2 + cos B) : 1/(2 + cos C)
Barycentrics    (sin A)/(2 + cos A) : :

Let A' be the midpoint of X(1) and the A-intouch point. Define B' and C' cyclically. The lines AA', BB', CC' concur in X(3296). (Randy Hutson, January 15, 2019)

X(3296) lies on these lines: 1,376   4,354   7,1058   21,999   65,1000   79,497   80,388   443,942   1039,1870

X(3296) = isogonal conjugate of X(3295)
X(3296) = cevapoint of X(1) and X(3338)
X(3296) = polar conjugate of isotomic conjugate of X(30679)
X(3296) = trilinear pole of line X(650)X(2423)

### X(3297) = INTERSECTION OF LINES X(485)X(496) AND X(486)X(495)

Trilinears        2 + sin A : 2 + sin B : 2 + sin C
Barycentrics   (2 + sin A)(sin A) : (2 + sin B)(sin B) : (2 + sin C)(sin C)

X(3297) lies on these lines: 1,6   371,999   485,496   486,495   605,1496   1151,2066

### X(3298) = INTERSECTION OF LINES X(485)X(495) AND X(486)X(496)

Trilinears        2 - sin A : 2 - sin B : 2 - sin C
Barycentrics   (2 - sin A)(sin A) : (2 - sin B)(sin B) : (2 - sin C)(sin C)

X(3298) lies on these lines: 1,6   372,999   485,495   486,496   606,1496   1151,2067

### X(3299) = INTERSECTION OF LINES X(35)X(372) AND X(36)X(371)

Trilinears        1 + 2 sin A : 1 + 2 sin B : 1 + 2 sin C
Trilinears        a + R : b +R : c + R
Barycentrics   (1 + 2 sin A)(sin A) : (1 + 2 sin B)(sin B) : (1 + 2 sin C)(sin C)

X(3299) lies on these lines: 1,6   35,372   36,371   42,589   46,1702   47,605   482,651

X(3299) = isogonal conjugate of X(3300)
X(3299) = {X(1),X(6)}-harmonic conjugate of X(3301)

### X(3300) = ISOGONAL CONJUGATE OF X(3299)

Trilinears        1/(1 + 2 sin A) : 1/(1 + 2 sin B) : 1/(1 + 2 sin C)
Barycentrics   (sin A)/(1 + 2 sin A) : (sin B)/(1 + 2 sin B) : (sin C)/(1 + 2 sin C)

X(3300) lies on these lines: 1,615   57,1373   498,1336   499,1123

X(3300) = isogonal conjugate of X(3299)

### X(3301) = INTERSECTION OF LINES X(35)X(371) AND X(36)X(372)

Trilinears    1 - 2 sin A : 1 - 2 sin B : 1 - 2 sin C
Trilinears    a - R : b -R : c - R
Barycentrics   (1 - 2 sin A)(sin A) : (1 - 2 sin B)(sin B) : (1 - 2 sin C)(sin C)

X(3301) lies on these lines: 1,6   35,371   36,372   42,588   46,1703   47,606   481,651

X(3301) = isogonal conjugate of X(3302)
X(3301) = {X(1),X(6)}-harmonic conjugate of X(3299)

### X(3302) = ISOGONAL CONJUGATE OF X(3301)

Trilinears        1/(1 - 2 sin A) : 1/(1 - 2 sin B) : 1/(1 - 2 sin C)
Barycentrics   (sin A)/(1 - 2 sin A) : (sin B)/(1 - 2 sin B) : (sin C)/(1 - 2 sin C)

X(3302) lies on these lines: 1,590   57,1374   498,1123   499,1336

X(3302) = isogonal conjugate of X(3301)

### X(3303) = INTERSECTION OF LINES X(1)X(3) AND X(496)X(498)

Trilinears        3 + cos A : 3 + cos B : 3 + cos C
Barycentrics   (sin A)(3 + cos A) :(sin B)(3 + cos B) : (sin C)(3 + cos C)

X(3303) lies on these lines: 1,3   8,344   11,1058   12,497   42,1191   145,958   198,1953   221,2293   377,528   388,390   474,551   495,546   496,498   499,632   500,1480   575,613   576,611   939,1411   943,1000   950,954   1193,1616   1201,2177

X(3303) = crosssum of X(1) and X(3333)

### X(3304) = INTERSECTION OF LINES X(1)X(3) AND X(495)X(499)

Trilinears        3 - cos A : 3 - cos B : 3 - cos C
Barycentrics   (sin A)(3 - cos A) :(sin B)(3 - cos B) : (sin C)(3 - cos C)

X(3304) lies on these lines: 1,3   7,1476   11,153   12,1056   31,1616   33,1398   106,386   144,1001   145,1376   198,1100   474,519   495,499   496,546   529,2478   575,611   576,613   1149,1191   1201,1696

X(3304) = midpoint of X(1) and X(3338)
X(3304) = crosssum of X(1) and X(1697)

### X(3305) = INTERSECTION OF LINES X(1)X(748) AND X(2)X(7)

Trilinears        (2 + cos A) csc A : (2 + cos B) csc B : (2 + cos C) csc C
Barycentrics   2 + cos A : 2 + cos B : 2 + cos C

X(3305) lies on these lines: 1,748   2,7   10,1479   19,469   21,936   43,968   78,405   81,1743   169,857   200,1621   210,1001   306,344   614,984   750,1707   975,1724

### X(3306) = INTERSECTION OF LINES X(1)X(88) AND X(2)X(7)

Trilinears        (2 - cos A) csc A : (2 - cos B) csc B : (2 - cos C) csc C
Barycentrics   2 - cos A : 2 - cos B : 2 - cos C

X(3306) lies on these lines: 1,88   2,7   46,1125   77,1465   81,2999   85,658   92,1435   145,1706   165,1621   171,614   354,1376   377,1210   612,982   748,1707   940,1100   1001,1155   1038,1393

X(3306) = crossdifference of every pair of points on the line X(663)X(1635)

### X(3307) = ISOGONAL CONJUGATE OF X(1381)

Trilinears        1/x : 1/y : 1/z, where x : y : z = X(1381)
Barycentrics   a/x : b/y : c/z, where x : y : z = X(1381)

As the isogonal conjugate of a point on the circumcircle, X(3307) lies on the line at infinity.

The asymptotes of the Feuerbach hyperbola meet the infinity line in X(3307) and X(3308). See the note at X(2574).

X(3307) lies on these (parallel) lines: 11,2446   30,511   104,1382

X(3307) = isogonal conjugate of X(1381)

### X(3308) = ISOGONAL CONJUGATE OF X(1382)

Trilinears        1/x : 1/y : 1/z, where x : y : z = X(1382)
Barycentrics   a/x : b/y : c/z, where x : y : z = X(1382)

As the isogonal conjugate of a point on the circumcircle, X(3308) lies on the line at infinity.

X(3308) lies on these (parallel) lines: 11,2447   30,511   100, 1382   104, 1381

X(3308) = isogonal conjugate of X(1382)

### X(3309) = ISOGONAL CONJUGATE OF X(1292)

Trilinears    (b - c)(a2 + b2 + c2 - 2ab - 2ac) : :

As the isogonal conjugate of a point on the circumcircle, X(3309) lies on the line at infinity.

X(3309) lies on these (parallel) lines: 1,3669   3,667   4,885   30,511   74,2752   98,2711   99,2704   100,2742   101,2736   102,2751   103,2725   104,840   109,2730   110,2691   218,2440   644,1292   650,1734   663,905   764,1482   1293,2748   1294,2749   1295,2750   1296,2753   1297,2754

X(3309) = isogonal conjugate of X(1292)
X(3309) = ideal point of PU(44)
X(3309) = bicentric difference of PU(44)
X(3309) = Thomson-isogonal conjugate of X(105)
X(3309) = Lucas-isogonal conjugate of X(105)
X(3309) = Cundy-Parry Psi transform of X(14267)

Barycentric Products of Perpendicular Directions

Suppose P and U are points on the line at infinity, given in barycentric coordinates by P = p : q : r and U = u : v : w. We call P and U perpendicular directions if for every point X not in the line at infinity, the lines XP and XU are perpendicular. (It is well known that if P and U are an antipodal pair on the circumcircle, then their isogonal conjugates are perpendicular directions.)

Theorem: If P and U are perpendicular directions, then their barycentric product lies on the orthic axis.

An outline of a proof follows. For given P on the line at infinity, the direction perpendicular to P is given by

a/d1 : b/d2 : c/d3,

where

d1 = a2qr cos B cos C - p(b2r + c2q) cos A
d2 = b2rp cos C cos A - q(c2p + a2r) cos B
d3 = c2pq cos A cos B - r(a2q + b2p) cos C,

and the barycentric product P*U is the point x : y : z given by

ap/d1 : bq/d2 : cr/d3.

For P to lie on the line at infinity means that r = - p - q. Substitution for r, a computer quickly shows that

x cot A + y cot B + z cot C = 0,

as desired.

The orthic axis is perpendicular to the Euler line. For this and other properties, visit MathWorld. The appearance of (I, J, K) in the following list means that X(K) is the barycentric product of perpendicular directions X(i) and X(j).

(30,1511,1637)
(511,523,2491)
(514,516,676)
(2574,2575,647)

(513,517,3310)

### X(3310) = BARYCENTRIC PRODUCT X(513)*X(517)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b - c)( -1 + cos B + cos C)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3310) lies on these lines: 6,654   37,1639   42,926   230,231   244,665   649,854   909,2423   1024,1945

X(3310) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,3259), (1411,3271), (2397,517), (2401,513), (2427,2183)
X(3310) = crosspoint of X(i) and X(j) for these (i,j): (2,901), (106,1461), (513,2401), (517,2397), (650,2432), (2183,2427)
X(3310) = crosssum of X(i) and X(j) for these (i,j): (6,900), (100,2427), (104,2423), (519,3239), (651,2406)
X(3310) = crossdifference of every pair of points on the line X(3)X(8)
X(3310) = perspector of hyperbola {A,B,C,X(4),X(56)} (circumconic centered at X(3259) and the isogonal conjugate of line X(3)X(8))
X(3310) = center of circumconic that is locus of trilinear poles of lines passing through X(3259)
X(3310) = intersection of trilinear polars of X(4) and X(56)
X(3310) = barycentric product of circumcircle intercepts of Sherman line

### X(3311) = INTERSECTION OF LINES X(3)X(6) AND X(4)X(1131)

Trilinears    cos A + 2 sin A : cos B + 2 sin B : cos C + 2 sin C
Trilinears    cos(A - t) : cos(B - t) : cos(C - t), where t = arctan(2)
Trilinears    a + R cos A : :
Trilinears    a(SA + 2S) : :
Trilinears    a(b^2 + c^2 - a^2 + 4S) : :
Barycentrics    (cos A + 2 sin A)(sin A) : (cos B + 2 sin B)(sin B) : (cos C + 2 sin C)(sin C)
X(3311) = La'/Ra' + Lb'/Rb' + Lc'/Rc', where La', Lb', Lc' are the centers of the secondary Lucas circles, and Ra', Rb', Rc' are their radii

X(3311) is the perspector of each of the following pairs of triangles:
Lucas central triangle and the symmedial triangle (the cevian triangle of X(6))
Lucas tangents triangle and the Lucas(-1:1) central triangle
Lucas(2:3) central triangle and the circumsymmedial triangle.
Moreover, X(3311), is the radical center of the Lucas(4:1) circles. See X(371) and X(3312). (Randy Hutson, 9/23/2011)

X(3311) is the perspector of each of the pair of the following four triangles:
symmedial triangle. Lucas central triangle, Lucas(-1) secondary central triangle, 1st Lucas(-1) secondary tangents triangle. (Randy Hutson, September 14,. 2016)

X(3311) lies on these lines: 3,6   4,1131   5,1588   25,588   30,1587   381,485   486,590   488,1992   517,1702   591,641   640,1991   999,1124

X(3311) = isogonal conjugate of X(3316)
X(3311) = inverse-in-Brocard-circle of X(3312)
X(3311) = {X(6),X(371)}-harmonic conjugate of X(3)
X(3311) = {X(6),X(1151)}-harmonic conjugate of X(372)
X(3311) = {X(61),X(62)}-harmonic conjugate of X(3592)
X(3311) = {X(371),X(372)}-harmonic conjugate of X(1151)
X(3311) = {X(372),X(1151)}-harmonic conjugate of X(3)

### X(3312) = INTERSECTION OF LINES X(3)X(6) AND X(4)X(1132)

Trilinears    cos A - 2 sin A : cos B - 2 sin B : cos C - 2 sin C
Trilinears    cos(A + t) : cos(B + t) : cos(C + t), where t = arctan(2)
Barycentrics    (cos A - 2 sin A)(sin A) : (cos B - 2 sin B)(sin B) : (cos C - 2 sin C)(sin C)

X(3312) is the perspector of each of the following pairs of triangles:
Lucas(-1:1) central triangle and the symmedial triangle
Lucas(-1:1) tangents triangle and the Lucas central triangle
Lucas(-2:3) central triangle and the circumsymmedial triangle.
Moreover, X(3311), is the radical center of the Lucas(-4:1) circles. See X(371) and X(3311). (Randy Hutson, 9/23/2011)

X(3312) is the perspector of each of the pair of the following four triangles:
symmedial triangle, Lucas(-1) central triangle, Lucas secondary central triangle, 1st Lucas secondary tangents triangle (Randy Hutson, September 14, 2016)

X(3312) lies on these lines: 3,6   4,1132   5,1587   25,589   30,1588   381,486   487,1992   517,1703   591,639   615,1656   642,1991   999,1335

X(3312) = isogonal conjugate of X(3317)
X(3312) = inverse-in-Brocard-circle of X(3311)
X(3312) = {X(6),X(372)}-harmonic conjugate of X(3)
X(3312) = {X(6),X(1152)}-harmonic conjugate of X(371)
X(3312) = {X(61),X(62)}-harmonic conjugate of X(3594)
X(3312) = {X(371),X(1152)}-harmonic conjugate of X(3)
X(3312) = {X(371),X(372)}-harmonic conjugate of X(1152)

### X(3313) = INTERSECTION OF LINES X(3)X(6) AND X(66)X(69)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2)(b4 + c4 - a4)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3313) lies on these lines: 3,6   22,206   66,69   141,427   159,394   1205,2854   1216,1352

X(3313) = reflection of X(i) in X(j) for these (i,j): (52,182), (1352,1216), (1843,141)
X(3313) = X(69)-Ceva conjugate of X(141)
X(3313) = crosspoint of X(22) and X(315)
X(3313) = crosssum of X(66) and X(2353)

### X(3314) = INTERSECTION OF LINES X(2)X(6) AND X(76)X(115)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[(b2 + c2)2 - b2c2]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3314) lies on these lines: 2,6   3,147   76,115   315,384   982,2887

X(3314) = isotomic conjugate of X(3407)
X(3314) = crosspoint of X(327) and X(1502)

### X(3315) = INTERSECTION OF LINES X(1)X(88) AND X(105)X(110)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b)(a - c) + 2(b - c)2
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3315) lies on these lines: 1,88   2,1280   105,110   149,1086   291,1255   651,1421   982,1621   1283,1623

X(3315) = crosssum of X(678) and X(2087)

### X(3316) = ISOGONAL CONJUGATE OF X(3311)

Trilinears        1/(cos A + 2 sin A) : 1/(cos B + 2 sin B) : 1/(cos C + 2 sin C)
Trilinears        sec(A - t) : sec(B - t) : sec(C - t), where t = arctan(2)
Barycentrics   (sin A)/(cos A + 2 sin A) : (sin B)/(cos B + 2 sin B) : (sin C) /(cos C + 2 sin C)

X(3316) lies on these lines: 3,1131   4,590   5,1132   226,1374   376,1327   485,631

X(3316) = isogonal conjugate of X(3311)
X(3316) = X(1587)-cross conjugate of X(4)

### X(3317) = ISOGONAL CONJUGATE OF X(3312)

Trilinears        1/(cos A - 2 sin A) : 1/(cos B - 2 sin B) : 1/(cos C - 2 sin C)
Trilinears        sec(A + t) : sec(B + t) : sec(C + t), where t = arctan(2)
Barycentrics   (sin A)/(cos A - 2 sin A) : (sin B)/(cos B - 2 sin B) : (sin C) /(cos C - 2 sin C)

X(3317) lies on these lines: 3,1132   4,615   5,1131   226,1373   376,1328   486,631

X(3317) = isogonal conjugate of X(3312)
X(3317) = X(1588)-cross conjugate of X(4)

Points on the Incircle

Suppose that U = u : v : w (trilinears) is a point other than the symmedian point, X(6). Then the incircle transform of U is the point

T(U) = a(bw - cv)2/(b + c - a) : b(cu - aw)2/(c + a - b) : c(av - bu)2/(a + b - c),

which lies on the incircle. Geometrically, if U is not X(1) then T(U) is the perspector of ABC and the reflection of the intouch triangle in the line UX(1). If X is a point other than U and X(6), then T(X) = T(U) if and only if X lies on the line UX(6). The transform T is clearly closely related to the Brisse transform, described just before X(1354).

### X(3318) = INCIRCLE TRANSFORM T(X(603))

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(bw - cv)2/(b + c - a) : b(cu - aw)2/(c + a - b) : c(av - bu)2/(a + b - c), where u : v : w = X(603)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b + c - a)(b - c)2[a3 + a2(b + c) - a(b + c)2 - (b + c)(b - c)2]2      (E. Danneels)

X(3318) lies on the incircle and these lines: 1,1359   11,123   55,108   56,1295   944,1317

X(3318) = reflection of X(1359) in X(1)

X(3318) = X(108)-of-Mandart-incircle-triangle
X(3318) = homothetic center of intangents triangle and reflection of extangents triangle in X(108)

### X(3319) = INCIRCLE TRANSFORM T(X(654))

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(bw - cv)2/(b + c - a) : b(cu - aw)2/(c + a - b) : c(av - bu)2/(a + b - c), where u : v : w = X(654)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = [2a4 - 2a3(b + c) - a2(b2 - 4bc + c2) + 2a(b + c)(b - c)2 - (b2 - c2)2]2/(b + c - a)      (E. Danneels)

X(3319) lies on the incircle and these lines: 1,3326   11,515   55,2716   56,2222   513,1361   517,1364   522,1317   944,2720

X(3319) = reflection of X(3326) in X(1)
X(3319) = anticomplement of X(5249)

### X(3320) = INCIRCLE TRANSFORM T(X(656))

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(bw - cv)2/(b + c - a) : b(cu - aw)2/(c + a - b) : c(av - bu)2/(a + b - c), where u : v : w = X(656)
Trilinears        g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a(b + c)2[a4 - a2bc - (b - c)2(b2 + bc + c2]2/(b + c - a)      (E. Danneels)
Barycentrics   ag(a,b,c) : bg(b,c,a) : cg(c,a,b)

X(3320) lies on the incircle and these lines: 11,132   12,127   55,1297   56,112   65,1367

X(3320) = X(1297)-of-Mandart-incircle-triangle
X(3320) = homothetic center of intangents triangle and reflection of extangents triangle in X(1297)

### X(3321) = INCIRCLE TRANSFORM T(X(663))

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(bw - cv)2/(b + c - a) : b(cu - aw)2/(c + a - b) : c(av - bu)2/(a + b - c), where u : v : w = X(663)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = [2a2 - a(b + c) - (b - c)2]2/(b + c - a)      (E. Danneels)

X(3321) lies on the incircle and these lines: 7,11   55,934   57,1358   1122,1357   1155,1323

X(3321) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,1323), 658,1638)
X(3321) = crosspoint of X(7) and X(1323)

### X(3322) = INCIRCLE TRANSFORM T(X(665))

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(bw - cv)2/(b + c - a) : b(cu - aw)2/(c + a - b) : c(av - bu)2/(a + b - c), where u : v : w = X(665)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = [2a3 - 2a2(b + c) + a(b2 + c2) - (b + c)(b - c)2]2/(b + c - a)      (E. Danneels)

X(3322) lies on the incircle and these lines: 7,840   55,2222   56,1308   513,1362   514,1317   516,1155   1283,1284

X(3322) = reflection of X(3328) in X(1)

### X(3323) = INCIRCLE TRANSFORM T(X(692))

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(bw - cv)2/(b + c - a) : b(cu - aw)2/(c + a - b) : c(av - bu)2/(a + b - c), where u : v : w = X(692)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b - c)2(b2 + c2 - ab - ac)2/(b + c - a)      (E. Danneels)

X(3323) lies on the incircle and these lines: 7,840   55,2736   56,2725   348,1083   514,1358   518,1362

X(3323) = trilinear product of PU(154)

### X(3324) = INCIRCLE TRANSFORM T(X(822))

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(bw - cv)2/(b + c - a) : b(cu - aw)2/(c + a - b) : c(av - bu)2/(a + b - c), where u : v : w = X(822)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b + c)2[a6 - a4(2b2 - 3bc + 2c2) + a2(b - c)2(b2 + c2) - bc(b2 - c2)2]2/(b + c - a)      (E. Danneels)

X(3324) lies on the incircle and these lines: 12,122   55,1294   56,107   65,1363   133,1838

X(3324) = X(1294)-of-Mandart-incircle-triangle
X(3324) = homothetic center of intangents triangle and reflection of extangents triangle in X(1294)

### X(3325) = INCIRCLE TRANSFORM T(X(896))

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(bw - cv)2/(b + c - a) : b(cu - aw)2/(c + a - b) : c(av - bu)2/(a + b - c), where u : v : w = X(896)
Trilinears        g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a(b - c)2(a2 + b2 + c2 + 3bc)2/(b + c - a)      (E. Danneels)
Barycentrics   ag(a,b,c) : bg(b,c,a) : cg(c,a,b)

X(3325) lies on the incircle and these lines: 12,126   55,1296   56,111   65,1366

X(3325) = X(1296)-of-Mandart-incircle-triangle
X(3325) = homothetic center of intangents triangle and reflection of extangents triangle in X(1296)

### X(3326) = INCIRCLE TRANSFORM T(X(909))

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(bw - cv)2/(b + c - a) : b(cu - aw)2/(c + a - b) : c(av - bu)2/(a + b - c), where u : v : w = X(909)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b + c - a)(b - c)2[a2(b + c) - 2abc - (b + c)(b - c)2]2      (E. Danneels)

X(3326) lies on the incircle and these lines: 1,3319   36,1354   55,2222   56,2716   125,2618   513,1364   517,1361   1118,2745

X(3326) = reflection of X(3319) in X(1)
X(3326) = contact point of incircle and the Sherman line (3259,3326) (see http://forumgeom.fau.edu/FG2012volume12/FG201220.pdf)

### X(3327) = INCIRCLE TRANSFORM T(X(2290))

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(bw - cv)2/(b + c - a) : b(cu - aw)2/(c + a - b) : c(av - bu)2/(a + b - c), where u : v : w = X(2290)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3327) lies on the incircle and these lines: 12,128   55,930   56,1141   496,1263

### X(3328) = INCIRCLE TRANSFORM T(X(2291))

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(bw - cv)2/(b + c - a) : b(cu - aw)2/(c + a - b) : c(av - bu)2/(a + b - c), where u : v : w = X(2291)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3328) lies on the incircle and these lines: 1,3322   55,1308   56,2717   516,1317   517,1362

X(3328) = reflection of X(i) in X(j) for these (i,j): (1155,1323), (3322,1)
X(3328) = X(7)-Ceva conjugate of X(1638)
X(3328) = crosspoint of X(i) in X(j) for these (i,j): (7,1638), (514,1323)

X(3328) = trilinear pole wrt intouch triangle of line X(7)X(11)

### X(3329) = INTERSECTION OF LINES X(2)X(6) AND X(39)X(83)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(a4 + 2a2b2 + 2a2c2 + b2c2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3329) lies on these lines: 2,6   5,147   39,83   98,575   182,262   427,1031

X(3229) = complement of X(3978) X(3229) = intersection of line X(2)X(39)[X(194)] and line through X(2)-Ceva conjugate of X(194) and X(194)-Ceva conjugate of X(2) X(3229) = crossdifference of PU(148)

### X(3330) = INTERSECTION OF LINES X(4)X(6) AND X(44)X(513)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos C)(sin 2A - sin 2B) - (cos B)(sin 2C - sin 2A)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3330) lies on these lines: 4,6   9,1745   37,73   44,513   198,408   216,1765   651,857   1100,2654

X(3330) = X(1294)-Ceva conjugate of X(55)
X(3330) = crosssum of X(1) and X(3330)
X(3330) = crossdifference of every pair of points on the line X(1)X(520)

### X(3331) = INTERSECTION OF LINES X(4)X(6) AND X(187)X(237)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc B cos C)(sin 2A - sin 2B) - (csc C cos B)(sin 2C - sin 2A)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3331) lies on these lines: 4,6   30,1625   74,1987   111,2713   112,1971   187,237

X(3331) = reflection of X(i) in X(j) for these (i,j): (3269,232), (3289,1625)
X(3331) = crossdifference of every pair of points on the line X(2)X(520)

### X(3332) = INTERSECTION OF LINES X(1)X(7) AND X(4)X(6)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos C)(1 + cos B)(cos C - cos A)(sin 2A - sin 2B) - (cos B)(1 + cos C)(cos A - cos B)(sin 2C - sin 2A)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3332) lies on these lines: 1,7   2,1754   4,6   42,2947   219,2550   278,1456

X(3332) = crossdifference of every pair of points on the line X(520)X(657)

### X(3333) = POHOATA POINT

Trilinears    (a - b + c)(a + b - c)(a + b + c) + 4abc : :

Let I be the incenter, X(1), and let KA be the symmedian point of triangle IBC; define KB, KC cyclically. Let X be the midpoint of segment AI,and define Y, Z cyclically. Then the triangles KAKBKC and XYZ are perspective, and their perspector is X(3333). Contributed by Cosmin Pohoata, April 4, 2008.

Let R be the circumradius, r the inradius, and rA the radius of the A-excircle. The trilinears given above are equivalent to 2R - rA : 2R - rB : 2R - rC. The 1st trilinear representation

(a - b + c)(a + b - c)(a + b + c) + 4abc

shows that X(3333) lies on the line IO of the incenter and the circumcenter, which is also the line X(1)X(57); viz., X(57) has 1st trilinear (a - b + c)(a + b - c) and X(1) has 1st trilinear 1, so that the above representation may be viewed as a linear combination using as coefficients the symmetric functions a + b + c and 4abc. The representation generalizes to

(a - b + c)(a + b - c)(a + b + c) + kabc,

where k is a symmetric function of degree 0 in a,b,c. Every point on the line IO necessary has such a form. (Indeed, every point on every line is given by such linear combinations of any two points on the line.) For the line IO, Peter Moses contributed (April 24, 2008) a list of points with corresponding functions k(a,b,c). Aside from inradius r and circumradius R, other symbols used are identified elsewhere in ETC as indicated, or else are formulated here: s = (a+b+c)/2; J as at X(1113); e as at X(1340); SA, SB SC as at X(1117); and sA = (-a+b+c)/2, sB = (a-b+c), sC = (a+b-c)/2.

Xk
1(k infinite)
3k = -2(r + 2R)/R
35k = -2(r + 3R)/R
36k = -2(r + R)/R
40k = -4
46k = -2
55k = -2(r + 4R)/R
56k = -2r/R
57k = 0
65k = 2r/R
165k = -(r +4r)/R
171k = -(r2 + 4rR - s2)/(rR)
241k = -2s2/(rR + 4R2)
354k = 2(r + 4R)//R
484k = -3
559k = 2(31/2s//R
940k = (a + b + c)3/(abc)
942k = 2(r + 2R)/R
980k = -2s2(r2 + s2)/(rR(r2 + 4rR + s2))
982k = -(a + b + c)(a2 + b2 + c2)/(abc)
986k = (r2 - s2)/(rR)
988k = -(r2 + s2)/(rR)
999k = -2(r - 2rR)/R
1038k = (J2 - 4r/R - 9)/2
1040k = - (J2 + 4r/R + 7)/2
1060k = J2 - 2r/R - 5
1062k = - (J2 + 2r/R + 3)
1082k = -2(31/2)s/R
1115k = -2(r + 4R)/(3R)
1159k = 2(5r + 2R)/(3R)
1214k = -2s2/(rR + 2R2)
1319k = -6r/R
1381k = 2(-r - R + (R2 - 2rR)1/2)/R
1382k = 2(-r - R + -(R2 - 2rR)1/2)/R
1385k = -2(3r + 2R)/R
1388k = -10r/R
1402k = -2r(r2 + 4rR + s2)/(R(r2 + s2))
1403k = -2r(r2 + 4rR + s2)/(R(r2 - 4rR + s2))
1420k = -4r/R
1429k = -2r(a + b + c)2/(R(a2 + b2 + c2))
1454k = -4r/(2r + R)
1460k = 2(area)(a2 + b2 + c2)/(2Rs(r2 - s2))
1466k = -2r(r + 2R)/(rR - 2R2)
1467k = -r(r + 2R)/R2
1470k = 2r(r + R)/(R2 - rR)
1482k = 2(3r - 2R)/R
1617k = -2r(r + 4R)/(2R2 + rR)
1697k = -8
1735k = -2 + (J2 - 1)R/(2r)
1754k = -(4r + (7 + J2)R)/(2r + 4R2)
1758k = -(r2 + 4rR + s2)/(2rR + R2)
1764k = -4s2/(r2 + s2)
1771k = -2 + (1 - J2)R/(2r)
1936k = [s2 - (r + 2R)(4 + 4R)]/(Rr + R2)
2077k = - 4 + 2r2 /(R2 - rR)
2078k = -2(r2 + 4rR)/(rR + R2)
2093k = (r - 2R)/R
2095k = (2r2 - 8R2)/(3rR + 2R2)
2098k = 2(3r - 4R)/R
2099k = 6r/R
2446k = -4(1 + (1 - 2r/R)1/2) + 2r/R
2447k = -4(1 - (1 - 2r/R)1/2) + 2r/R
2448k = -3 + (1 + 2r/R)1/2
2449k = -3 - (1 + 2r/R)1/2
2556k = -4 + 2r(e + E)/(Re - RE), where E = (1 - 2r/R)1/2
2557k = -4 + 2r(e - E)/(Re + RE), where E = (1 - 2r/R)1/2
2464k = -4(1 + e) - 2r/R
2465k = -4(1 - e) - 2r/R
2572k = -4 - 2r/(R + eR)
2573k = -4 - 2r/(R - eR)
2646k = -2(3r + 4R)/R
3057k = 2(r - 4R)/R
3072k = (R - RJ2 -4r)/(2r + 2R)
3075k = (SASBSC2)/(abcsAsBsC2)
3245k = -2(5R - r)/(3R)
3256k = 2r(r + 4R)/(R - rR2)
3303k = -2(r + 8R)/R
3304k = -2(r - 4R)/R
3333k = 4
3336k = -1
3337k = 1
3338k = 2
3339k = r/R
3340k = 4r/R
3361k = -r/R

Let A' be the midpoint of X(1) and A-intouch point. Define B' and C' cyclically. Traingle A'B'C' is homothetic to the excentral triangle, and the center of homothety is X(3333). (Randy Hutson, December 2, 2017)

If you have The Geometer's Sketchpad, you can view Pohoata Point.

X(3333) lies on these lines:

1,3   4,1435   7,84   9,1125   10,1056   48,1449   58,2191   81,2360   109,1497   200,474   226,3086   386,1066   388,1210   495,1698   496,1699   515,938   516,1058   518,936   519,1706   580,1471   581,1458   614,1453   1106,2263   1203,3157   1387,1768   1731,2257   2136,3244

X(3333) = midpoint of X(1) and X(3339)
X(3333) = crosssum of X(55) and X(2256)
X(3333) = X(1598)-of-excentral-triangle

### X(3334) = PERSPECTOR OF TRIANGLES T1T2T3 AND J1J2J3

Trilinears         f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos A + 2 sin(B/3) sin(C/3) + sin(A/3 +π/6)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

The equilateral triangles T1T2T3 and J1J2J3 are defined just before X(3272). The points X(3334) and X(3335), contributed April 16, 2008 by Peter Moses, complete a list (just before X(3272) of perspectors associated with equilateral triangles.

X(3334) lies on this line: 16,358

### X(3335) = PERSPECTOR OF TRIANGLES N1N2N3 AND J1J2J3

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos A - 2 sin(B/3) sin(C/3) - sin(A/3 +π/6)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

The equilateral triangles N1N2N3 and J1J2J3 are defined just before X(3272).

### X(3336) = INTERSECTION OF LINES X(1)X(3) AND X(7)X(498)

Trilinears    (a - b + c)(a + b - c)(a + b + c) - abc : :
Trilinears    1 + 2 (cos B + cos C - cos A) : :

X(3336) lies on the Napoleon cubic and these lines:

1,3   2,191   4,1768   5,79   7,498   17,1652   18,1653   54,3468   58,1325   62,2306   63,1698   109,1393   244,595   269,1079   404,758   412,1784   579,1781   583,2160   920,1445   1046,1054   1158,1699   1210,1770   1254,1772   1478,1788   1707,1775   1727,1836   3459,3460   3461,3462

X(3336) = X(i)-Ceva conjugate of X(j) for these (i,j): (5,3468), (77,1745)
X(3336) = {X(1)X(46)}-harmonic conjugate of X(484)
X(3336) = SS(A->3A) of X(1507)
X(3336) = isogonal conjugate of X(3467)
X(3336) = Kosnita(X(484),X(3)) point

### X(3337) = INTERSECTION OF LINES X(1)X(3) AND X(7)X(499)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b + c)(a + b - c)(a + b + c) + abc
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3337) lies on these lines:

1,3   7,499   11,79   47,1471   58,229   81,501   202,2306   553,1776   946,1768   1111,1434   1399,1421   1731,1781

### X(3338) = INTERSECTION OF LINES X(1)X(3) AND X(7)X(90)

Trilinears    (a - b + c)(a + b - c)(a + b + c) + 2abc ::
Trilinears    2 s (b c - b s - c s + s^2) + R*S : :
Trilinears    ra + R : :, where ra, rb, rc are the exradii

X(3338) lies on these lines: 1,3   7,90   9,583   38,975   58,614   63,1125   84,1699   169,1475   226,499   255,1471   388,1737   474,518   496,1836   497,1770   553,946   584,1449   990,1717   1056,1788   1106,1448   1210,1478   1398,1905   1435,1838   1452,1870   1468,1718   1537,1768   1723,2260

X(3338) = reflection of X(1) in X(3304)
X(3338) = X(3296)-Ceva conjugate of X(1)
X(3338) = {X(1),X(57)}-harmonic conjugate of X(46)

### X(3339) = INTERSECTION OF LINES X(1)X(3) AND X(7)X(10)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b + c)(a + b - c)(a + b + c) + (r/R)abc
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

See the note at X(3333).

X(3339) lies on these lines:

1,3   7,10   109,1451   169,1046   196,1838   208,1844   221,1203   223,2939   269,1126   386,1042   388,533   516,938   518,1706   527,2551   595,1471   758,936   959,978   1210,1699   1406,2003   1698,1788   1721,2955   1767,2956

X(3339) = reflection of X(1) in X(3333)

### X(3340) = INTERSECTION OF LINES X(1)X(3) AND X(7)X(145)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b + c)(a + b - c)(a + b + c) + (4r/R)abc
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

See the note at X(3333).

The reflection of the excentral triangle in X(1) and the intouch triangle are homothetic from X(3340. (Randy Hutson, January 29, 2015)

X(3340) lies on these lines:

1,3   7,145   8,226   9,1405   34,1126   78,1706   84,1389   109,1468   221,2003   278,1869   386,1457,   388,519   595,1451   950,962   1125,1788   1404,1449   1419,2263   1871,1887   1953,2257

X(3340) = reflection of X(1697) in X(1)
X(3340) = 2nd-extouch-to-intouch similarity image of X(8)

### X(3341) = X(2)-CEVA CONJUGATE OF X(282)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = x(-ax + by + cz), where x : y : z = X(282)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3341) lies on the Thomson cubic and these lines: 1,1073   2,271   3,3352   4,57   6,282   9,3344   223,3349   280,938

X(3341) = isogonal conjugate of X(3342)
X(3341) = complement of X(1034)
X(3341) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,282), (271,84)
X(3341) = X(i)-cross conjugate of X(j) for these (i,j): (6,3352), (1035,1490)

### X(3342) = ISOGONAL CONJUGATE OF X(3341)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/(x(-ax + by + cz)), where x : y : z = X(282)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3342) lies on the Thomson cubic and these lines: 1,3343   2,271   3,223   6,3351   9,1249   57,3350

X(3342) = isogonal conjugate of X(3341)
X(3342) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,3351), (1034,3345)
X(3342) = X(i)-cross conjugate of X(j) for these (i,j): (6,223), (208,40)

### X(3343) = X(2)-CEVA CONJUGATE OF X(1073)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = x(-ax + by + cz), where x : y : z = X(1073)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3343) lies on the Thomson cubic and these lines: 1,3342   2,1032   3,2130   4,64   6,1073   9,3352   57,282   154,1301   1249,3356

X(3343) = isogonal conjugate of X(3344)
X(3343) = complement of X(1032)
X(3343) = X(2)-Ceva conjugate of X(1073)
X(3343) = X(i)-cross conjugate of X(j) for these (i,j): (6,3349), (1033,1498)

### X(3344) = ISOGONAL CONJUGATE OF X(3343)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/(x(-ax + by + cz)), where x : y : z = X(1073)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3344) lies on the Thomson cubic and these lines: 1,3351   2,1032   3,1033   6,3350   9,3341

X(3344) = isogonal conjugate of X(3343)
X(3344) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,3350), (1032,3346)
X(3344) = X(6)-cross conjugate of X(1249)

### X(3345) = ISOGONAL CONJUGATE OF X(1490)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(1 + sec A - sec B - sec C)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3345) lies on the Darboux cubic and these lines: 1,196   3,223   4,282   20,78   40,219   64,3472   84,2130   165,1794   283,1817   1722,2636   1753,2338

X(3345) = reflection of X(3182) in X(3)
X(3345) = isogonal conjugate of X(1490)
X(3345) = X(1034)-Ceva conjugate of X(3342)
X(3345) = cevapoint of X(649) and X(2638)
X(3345) = X(i)-cross conjugate of X(j) for these (i,j): (34,1), (64,84), (1436,57)

### X(3346) = ISOGONAL CONJUGATE OF X(1498)

Trilinears    1/[(sin A)(tan2A - tan2B - tan2C)] : :

Let A'B'C' be the cevian triangle of X(20). Let A" be the orthocenter of AB'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(3346). (Randy Hutson, November 18, 2015)

Let M be a point on the slideline BC of triangle ABC. Let (AM) be the circle with diameter AM. Let Mb be the point, other than A, in (AM)∩AC, and let Mc be the point, other than A, in (AM)∩AB. Let Tb be the line tangent to (AM) at Mb, and let Tc be the line tangent to (AM) at Mc. As M varies on BC, the locus of Tb∩Tc is a line, La. Define Lb and Lc cyclically. Let A' = Lb∩Lc, and define B' and C' cyclically. The triangle A'B'C' is perspective to ABCF, and the perspector is X(3346). (Angel Montesdeoca, November 11, 2017)

X(3346) lies on the Darboux cubic and these lines: 1,3353   3,1033   4,1073   20,394   40,3182   64,3355   84,3472

X(3346) = reflection of X(3183) in X(3)
X(3346) = isogonal conjugate of X(1498)
X(3346) = X(i)-Ceva conjugate of X(j) for these (i,j): (20,3355), (1032,3344)
X(3346) = cevapoint of X(122) and X(523)
X(3346) = X(i)-cross conjugate of X(j) for these (i,j): (64,4), (393,2)
X(3346) = intersection of tangents at X(20) and X(253) to Lucas cubic K007
X(3346) = perspector of ABC and the reflection in X(1249) of the antipedal triangle of X(1249)

### X(3347) = ISOGONAL CONJUGATE OF X(3182)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/x : 1/y : 1/z, where x : y : z = X(3182)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3347) lies on the Darboux cubic and these lines: 1,2130   3,3353   4,3472   20,3182   40,1712   84,3355   1490,1498

X(3347) = reflection of X(3353) in X(3)
X(3347) = isogonal conjugate of X(3182)
X(3347) = X(64)-cross conjugate of X(1)

### X(3348) = ISOGONAL CONJUGATE OF X(3183)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/x : 1/y : 1/z, where x : y : z = X(3183)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3348) lies on the Darboux cubic and these lines: 1,3472   3,2130   4,3355   20,3183   40,3353   1490,3182

X(3348) = reflection of X(2130) in X(3)
X(3348) = isogonal conjugate of X(3183)

### X(3349) = INTERSECTION OF LINES X(3)X(2130) AND X(4)X(2131)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[x + y + z - 2yz(y + z - x)/(y2 + z2 - x2)]- 1,
where x : y : z = 1/(b2 + c2 - a2) : 1/(c2 + a2 - b2) : 1/(a2 + b2 - c2)    (M. Iliev, 10/27/08)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3349) lies on the Thomson cubic and these lines: 3,2130   4,2131   9,3351   223,3341

X(3349) = isogonal conjugate of X(3350)
X(3349) = X(6)-cross conjugate of X(3343)

### X(3350) = INTERSECTION OF LINES X(3)X(2131) AND X(4)X(1073)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[x + y + z - 2yz(y + z - x)/(y2 + z2 - x2)],
where x : y : z = 1/(b2 + c2 - a2) : 1/(c2 + a2 - b2) : 1/(a2 + b2 - c2)    (M. Iliev, 10/27/08)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3350) lies on the Thomson cubic and these lines: 1,3352   2,3349   3,2131   4,1073   6,3344   57,3342

X(3350) = isogonal conjugate of X(3349)
X(3350) = X(2)-Ceva conjugate of X(3344)
X(3350) = X(6)-cross conjugate of X(3356)

### X(3351) = X(2)-CEVA CONJUGATE OF X(3342)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = x(-ax + by + cz), where x : y : z = X(3342)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3351) lies on the Thomson cubic and these lines: 1,3344   2,3352   4,282   57,1073   6,3342   9,3349   223,3356

X(3351) = isogonal conjugate of X(3352)
X(3351) = X(2)-Ceva conjugate of X(3342)

### X(3352) = ISOGONAL CONJUGATE OF X(3351)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/x(3351)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3352) lies on the Thomson cubic and these lines: 1,3350   2,3351   3,3341   9,3343

X(3352) = isogonal conjugate of X(3351)
X(3352) = X(6)-cross conjugate of X(3341)

### X(3353) = X(20)-CEVA CONJUGATE OF X(84)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = x(84)/(-x(84)/x(20) + y(84)/y(20) + z(84)/z(20))
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3353) lies on the Darboux cubic and these lines: 1,3346   3,3347   4,282   20,3354   40,3348   64,84   1490,2131   1498,3473

X(3353) = reflection of X(3347) in X(3)
X(3353) = isogonal conjugate of X(3354)
X(3353) = X(20)-Ceva conjugate of X(84)

### X(3354) = ISOGONAL CONJUGATE OF X(3353)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/x(3353)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3354) lies on the Darboux cubic and these lines: 1,3355   3,3472   20,3353   40,2130   1490,3183   1498,3182

X(3354) = reflection of X(3472) in X(3)
X(3354) = isogonal conjugate of X(3353)
X(3354) = X(64)-cross conjugate of X(40)

### X(3355) = REFLECTION OF X(2131) IN X(3)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2x3 - x2131 (using actual trilinear distances)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3355) lies on the Darboux cubic and these lines: 1,3354   3,2130   4,3348   40,3473   64,3346   84,3347

X(3355) = isogonal conjugate of X(3637)
X(3355) = reflection of X(2131) in X(3)
X(3355) = X(20)-Ceva conjugate of X(3346)

### X(3356) = X(6)-CROSS CONJUGATE OF X(3350)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = u/(-avw + bwu + cuv), where u : v : w = X(3350)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3356) lies on the Thomson cubic and these lines: 3,2131   223,3351   1249,3343

X(3356) = X(6)-cross conjugate of X(3350)

### X(3357) = MIDPOINT OF X(3) AND X(64)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = x3 + x64 (using actual trilinear distances)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(3357) = (1 + |OK|/R)*X(3) + 3X(6)

X(3357) lies on these lines:

3,64   4,74   20,2888   140,2883   185,378   195,2935   382,1853   389,1593   550,1503   576,2781   1092,2071   1192,1598

X(3357) = midpoint of X(3) and X(64)
X(3357) = reflection of X(2883) in X(140)
X(3357) = inverse-in-Jerabek-hyperbola of X(1204)
X(3357) = {X(4),X(74)}-harmonic conjugate of X(1204)
X(3357) = X(8)-of-Trinh-triangle if ABC is acute
X(3357) = Trinh-isotomic conjugate of X(3098)

### X(3358) = MIDPOINT OF X(9) AND X(84)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = x9 + x84 (using actual trilinear distances)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(3358) = (1 + |OK|/R)*X(3) - 3X(6)

X(3358) lies on these lines:

3,9   4,1445   11,57   516,1158   954,1071

X(3358) = midpoint of X(9) and X(84)

### X(3359) = MIDPOINT OF X(2096) AND X(3421)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b + c)(a + b - c)(a + b + c) + kabc, where k = -2(2R - r)/(R - r). (Peter Moses, January 11, 2011)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3359) lies on these lines:

1,3   2,1519   9,119   10,1158   63,2096   72,2057   84,355   200,912   997,2800   1103,3157   1452,1753   1722,3073   2550,3358

X(3359) = midpoint of X(2096) and X(3421)

### X(3360) = SS(a → bc) OF X(20)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[3b4c4 - c4a4 - a4b4 +2a2(a2b2c2 - b2c4 - b4c2)]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3360) lies on these lines:

1,2162   6,194   159,2076   1740,2176

X(3360) = X(1613)-Ceva conjugate of X(6)

### X(3361) = INTERSECTION OF LINES X(1)X(3) AND X(7)X(1125)

Trilinears    (3a + b + c)/(-a + b + c)

Let OA be the circle tangent to side BC at its midpoint and to the circumcircle on the side ob BC opposite A. Define OB and OC cyclically. Let A' be the insimilicanter of OB and OC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(3361). See the reference at X(1001).

Let Ja, Jb, Jc be the excenters and I the incenter. Let A' be the centroid of JbJcI, and define B' and C' cyclically. A'B'C' is also the cross-triangle of the excentral and 2nd circumperp triangles. A'B'C' is homothetic to the inverse-in-incircle triangle at X(3361). (Randy Hutson, July 31 2018)

X(3361) lies on these lines:

1,3   7,1125   28,1435   34,2163   58,269   200,404   222,1203   223,1451   386,1458   388,1698   610,2260   614,1448   936,1445   961,1722   978,1400   995,1042   1416,1472   1427,1453   1468,2999

X(3361) = isogonal conjugate of X(4866)
X(3361) = {X(56),X(57)}-harmonic conjugate of X(1)
X(3361) = {X(57),X(1420)}-harmonic conjugate of X(65)

### X(3362) = ISOGONAL CONJUGATE OF X(1745)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(sec B + sec C - sec A)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3362) lies on the McKay cubic and these lines:

1,1075   3,1745   46,296   158,2638   283,1816   580,1795   1069,1936   1794,2947

X(3362) = isogonal conjugate of X(1745)
X(3362) = cevapoint of X(650) and X(2638)
X(3362) = X(4)-cross conjugate of X(1)
X(3362) = trilinear product of PU(126)

### X(3363) = X(6) OF PEDAL TRIANGLE OF X(2)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(-4a4 + 5b4 + 5c4 - 14b2c2 - 5c2a2 - 5a2b2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

As a point on the Euler line, X(3363) has Shinagawa coefficients (2(E + F)2 + 3S2,9S2).

X(3363) is the symmedian point of the pedal triangle of the centroid. A second construction uses the midpoint X(597) of the symmedian point and centroid and the midpoint X(115) of the two Fermat points, X(13) and X(14); specifically, X(3363) is the point in which the line X(115)X(597) meets the Euler line. Contributed by Po-chieh Chen and Shao-cheng Liu, May 29, 2008.

If you have The Geometer's Sketchpad, you can view X(3363).

X(3363) lies on these lines: 2,3   115,597

### X(3364) = COS(A - 5π/12) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(A - 5π/12)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3364) and related points occupy Bernard Gibert's Table 38 concerning these three loci: Brocard axis, Kiepert hyperbola, and the cubic K457. Trilinear equations for these curves, of degrees 1,2,3, respectively, in the variables a,b,c, are given as follows:

Curve Trilinears
Brocard axis cos(A + t) : cos(B + t) : cos(C + t)
Kiepert hyperbola sec(A + t) : sec(B + t) : sec(C + t)
K457 tan(A + t) : tan(B + t) : tan(C + t)

X(3364) lies on these lines: 3,6   14,485   17,486   18,590   202,2067 nbsp;  203,1124

X(3364) = isogonal conjugate of X(3366)
X(3364) = inverse-in-Brocard-circle of X(3390)
X(3364) = X(18)-Ceva conjugate of X(3365)
X(3364) = X(3206)-cross conjugate of X(3365)

### X(3365) = SIN(A - 5π/12) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin(A - 5π/12)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3365) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3365) lies on these lines: 3,6   14,486   17,485   18,615   203,1335

X(3365) = isogonal conjugate of X(3367)
X(3365) = inverse-in-Brocard-circle of X(3389)
X(3365) = X(18)-Ceva conjugate of X(3364)
X(3365) = X(3206)-cross conjugate of X(3364)

### X(3366) = SEC(A - 5π/12) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec(A - 5π/12)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3366) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3366) lies on the Kiepert hyperbola and these lines: 5,15   14,371   17,372   18,590

X(3366) = isogonal conjugate of X(3364)
X(3366) = X(62)-cross conjugate of X(3367)

### X(3367) = CSC(A - 5π/12) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = csc(A - 5π/12)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3367) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3367) lies on the Kiepert hyperbola and these lines: 5,15   14,372   17,371   18,615

X(3367) = isogonal conjugate of X(3365)
X(3367) = X(62)-cross conjugate of X(3366)

### X(3368) = COS(A - 2π/5) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(A - 2π/5)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3368) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

If you have The Geometer's Sketchpad, you can view X(3368).

X(3368) lies on these lines: 3,6   4,1139

X(3368) = isogonal conjugate of X(3370)
X(3368) = inverse-in-Brocard-circle of X(3395)
X(3368) = X(3382)-Ceva conjugate of X(3369)

### X(3369) = SIN(A - 2π/5) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin(A - 2π/5)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3369) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

If you have The Geometer's Sketchpad, you can view X(3369).

X(3369) lies on these lines: 2,1139   3,6

X(3369) = isogonal conjugate of X(1140)
X(3369) = {X(371),X(372)}-harmonic conjugate of X(3395)
X(3369) = inverse-in-Brocard-circle of X(3396)
X(3369) = X(3382)-Ceva conjugate of X(3368)
X(3369) = crosssum of X(3393) and X(3394)

### X(3370) = SEC(A - 2π/5) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec(A - 2π/5)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3370) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

Let A' be the center of the regular pentagon AA2A3A4A5, with opposite orientation than ABC and such that B ∈ A2A3 and C ∈ A4A5. Build B' and C' cyclically. Then the lines AA', BB', CC' concur at X(3370). (César E. Lozada, May 14, 2019), Hyacinthos #29013).

If you have The Geometer's Sketchpad, you can view X(3370).

X(3370) lies on the Kiepert hyperbola and these lines: 3,1139   5,1140

X(3370) = isogonal conjugate of X(3368)
X(3370) = X(3380)-cross conjugate of X(1140)

### X(3371) = COS(A - 3π/8) POINT

Trilinears    sin(A + π8) : :
Trilinears    cos(A - 3π/8) : :

X(3371) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3371) lies on this line: 3,6

X(3371) = isogonal conjugate of X(3373)
X(3371) = inverse-in-Brocard-circle of X(3386)
X(3371) = X(486)-Ceva conjugate of X(3372)
X(3371) = insimilicenter of circumcircle and 2nd Kenmotu circle
X(3371) = {X(371),X(372)}-harmonic conjugate of X(3385)

### X(3372) = SIN(A - 3π/8) POINT

Trilinears    cos(A + π/8) : :
Trilinears    sin(A - 3π;/8)

X(3372) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3372) lies on this line: 3,6

X(3372) = isogonal conjugate of X(3374)
X(3372) = inverse-in-Brocard-circle of X(3385)
X(3372) = X(486)-Ceva conjugate of X(3371)
X(3372) = exsimilicenter of circumcircle and 2nd Kenmotu circle
X(3372) = {X(371),X(372)}-harmonic conjugate of X(3386)

### X(3373) = SEC(A - 3π/8) POINT

Trilinears    sec(A - 3π/8) : :
Trilinears    csc(A + p/8) : :

X(3373) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3373) lies on the Kiepert hyperbola and this line: 5,371

X(3373) = isogonal conjugate of X(3371)
X(3373) = X(372)-cross conjugate of X(3374)

### X(3374) = CSC(A - 3π/8) POINT

Trilinears    csc(A - 3π/8) : :
Trilinears    sec(A + p/8) : :

X(3374) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3374) lies on the Kiepert hyperbola and this line: 5,371

X(3374) = isogonal conjugate of X(3372)
X(3374) = X(372)-cross conjugate of X(3373)

### X(3375) = TAN(A - π/3) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = tan(A - π/3)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3375) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3375) lies on these lines: 17,1652   19,2290

X(3375) = isogonal conjugate of X(3376)
X(3375) = X(1095)-cross conjugate of X(1)
X(3375) = trilinear product of X(16) and X(17)

### X(3376) = COT(A - π/3) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cot(A - π/3)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3376) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3376) lies on these lines: 1,1095   14,484   19,2166

X(3376) = isogonal conjugate of X(3375)
X(3376) = trilinear product of X(14) and X(61)

### X(3377) = TAN(A - π/4) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = tan(A - π/4)
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (1 - tan A)/(1 + tan A)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3377) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3377) lies on these lines: 1,1805   19,91   46,485

X(3377) = isogonal conjugate of X(3378)
X(3377) = trilinear quotient X(372)/X(371)

### X(3378) = COT(A - π/4) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cot(A - π/4)
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (1 + tan A)/(1 - tan A)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3378) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3378) lies on these lines: 1,1806   19,91   46,486

X(3378) = isogonal conjugate of X(3377)
X(3378) = trilinear quotient X(371)/X(372)

### X(3379) = COS(A - π/5) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(A - π/5)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3379) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

If you have The Geometer's Sketchpad, you can view X(3379).

X(3379) lies on these lines: 3,6   5,1140

X(3379) = isogonal conjugate of X(3381)
X(3379) = inverse-in-Brocard-circle of X(3393)
X(3379) = {X(371),X(372)}-harmonic conjugate of X(3394)
X(3379) = X(1139)-Ceva conjugate of X(3380)

### X(3380) = SIN(A - π/5) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin(A - π/5)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3380) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

If you have The Geometer's Sketchpad, you can view X(3380).

X(3380) lies on this line: 3,6

X(3380) = isogonal conjugate of X(3382)
X(3380) = inverse-in-Brocard-circle of X(3394)
X(3380) = X(1139)-Ceva conjugate of X(3379)
X(3380) = crosspoint of X(1140) and X(3370)
X(3380) = crosssum of X(3368) and X(3369)

### X(3381) = SEC(A - π/5) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec(A - π/5)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3381) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

If you have The Geometer's Sketchpad, you can view X(3381).

X(3381) lies on the Kiepert hyperbola.

X(3381) = isogonal conjugate of X(3379)
X(3381) = X(3396)-cross conjugate of X(3382)

### X(3382) = CSC(A - π/5) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = csc(A - π/5)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3382) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

If you have The Geometer's Sketchpad, you can view X(3382).

X(3382) lies on the Kiepert hyperbola.

X(3382) = isogonal conjugate of X(3380)
X(3382) = cevapoint of X(3368) and X(3369)
X(3382) = X(3396)-cross conjugate of X(3381)

### X(3383) = TAN(A - π/6) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = tan(A - π/6)
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = cot(A + π/3)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3383) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3383) lies on these lines: 1,1094   13,484   19,2166

X(3383) = isogonal conjugate of X(3384)
X(3383) = trilinear product of X(13) and X(62)

### X(3384) = COT(A - π/6) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cot(A - π/6)
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = tan(A + π/3)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3384) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3384) lies on these lines: 18,1653   19,2290

X(3384) = isogonal conjugate of X(3383)
X(3384) = X(1094)-cross conjugate of X(1)
X(3384) = trilinear product of X(15) and X(18)

### X(3385) = COS(A - π/8) POINT

Trilinears    cos(A - π/8)

X(3385) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3385) lies on this line: 3,6

X(3385) = isogonal conjugate of X(3387)
X(3385) = inverse-in-Brocard-circle of X(3372)
X(3385) = X(485)-Ceva conjugate of X(3386)
X(3385) = insimilicenter of circumcircle and 1st Kenmotu circle
X(3385) = {X(371),X(372)}-harmonic conjugate of X(3371)

### X(3386) = SIN(A - π/8) POINT

Trilinears    sin(A - π/8) : :

X(3386) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3386) lies on this line: 3,6

X(3386) = isogonal conjugate of X(3388)
X(3386) = inverse-in-Brocard-circle of X(3371)
X(3386) = X(485)-Ceva conjugate of X(3385)
X(3386) = exsimilicenter of circumcircle and 1st Kenmotu circle
X(3386) = {X(371),X(372)}-harmonic conjugate of X(3372)

### X(3387) = SEC(A - π/8) POINT

Trilinears    sec(A - π/8) : :

X(3387) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3387) lies on the Kiepert hyperbola and this line: 5,372

X(3387) = isogonal conjugate of X(3385)
X(3387) = X(371)-cross conjugate of X(3388)

### X(3388) = CSC(A - π/8) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = csc(A - π/8)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3388) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3388) lies on the Kiepert hyperbola and this line: 5,372

X(3388) = isogonal conjugate of X(3386)
X(3388) = X(371)-cross conjugate of X(3387)

### X(3389) = COS(A - π/12) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(A - π/12)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3389) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3389) lies on these lines: 3,6   13,485   17,590   18,486   202,1124   203,2067

X(3389) = isogonal conjugate of X(3391)
X(3389) = inverse-in-Brocard-circle of X(3365)
X(3389) = X(17)-Ceva conjugate of X(3390)
X(3389) = X(3205)-cross conjugate of X(3390)

### X(3390) = SIN(A - π/12) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin(A - π/12)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3390) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3390) lies on these lines: 3,6   13,486   17,615   18,485   202,1335

X(3390) = isogonal conjugate of X(3392)
X(3390) = inverse-in-Brocard-circle of X(3364)
X(3390) = X(17)-Ceva conjugate of X(3389) X(3390) = X(3205)-cross conjugate of X(3389)

### X(3391) = SEC(A - π/12) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec(A - π/12)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3391) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3391) lies on the Kiepert hyperbola and these lines: 5,16   13,371   17,590   18,372 18,485   202,1335

X(3391) = isogonal conjugate of X(3389)
X(3391) = X(61)-cross conjugate of X(3392)

### X(3392) = CSC(A - π/12) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = csc(A - π/12)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3392) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3392) lies on the Kiepert hyperbola and these lines: 5,16   13,372   17,615   18,371 18,485   202,1335

X(3392) = isogonal conjugate of X(3390)
X(3392) = X(61)-cross conjugate of X(3391)

### X(3393) = COS(A + π/5) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(A + π/5)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3393) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

If you have The Geometer's Sketchpad, you can view X(3393).

X(3393) lies on these lines: 3,6   5,1139

X(3393) = isogonal conjugate of X(5401)
X(3393) = inverse-in-Brocard-circle of X(3379)
X(3393) = X(1140)-Ceva conjugate of X(3394)

### X(3394) = SIN(A + π/5) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin(A + π/5)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3394) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

If you have The Geometer's Sketchpad, you can view X(3394).

X(3394) lies on this line: 3,6

X(3394) = inverse-in-Brocard-circle of X(3380)
X(3394) = X(1140)-Ceva conjugate of X(3393)
X(3394) = crosspoint of X(1139) and X(3397)
X(3394) = crosssum of X(3395) and X(3396)

### X(3395) = COS(A + 2π/5) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(A + 2π/5)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3395) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

If you have The Geometer's Sketchpad, you can view X(3395).

X(3395) lies on these lines: 3,6   4,1140

X(3395) = isogonal conjugate of X(3397)
X(3395) = {X(371),X(372)}-harmonic conjugate of X(3369)
X(3395) = inverse-in-Brocard-circle of X(3368)

### X(3396) = SIN(A + 2π/5) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin(A + 2π/5)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3396) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

If you have The Geometer's Sketchpad, you can view X(3396).

X(3396) lies on this line: 2,1140   3,6

X(3396) = isogonal conjugate of X(1139)
X(3396) = inverse-in-Brocard-circle of X(3369)
X(3396) = crosssum of X(3379) and X(3380)
X(3396) = crosspoint of X(3381) and X(3382)
X(3396) = {X(371),X(372)}-harmonic conjugate of X(3368)

### X(3397) = SEC(A + 2π/5) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec(A + 2π/5)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3397) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

Let A' be the center of the regular pentagon AA2A3A4A5, with the same orientation than ABC and such that B ∈ A2A3 and C ∈ A4A5. Build B' and C' cyclically. Then the lines AA', BB', CC' concur at X(3397). (César E. Lozada, May 14, 2019, Hyacinthos #29013).

If you have The Geometer's Sketchpad, you can view X(3397).

X(3397) lies on the Kiepert hyperbola and this line: 2,1140

X(3397) = isogonal conjugate of X(3395)
X(3397) = X(3394)-cross conjugate of X(1139)

### X(3398) = COS(A - 2ω) POINT

Trilinears    cos(A - 2ω) : :
Trilinears    e2sin(A - ω) - sin(A + ω) : :

X(3398) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

Randy Huston, September 5, 2015, gives three constructions of X(3398):

(1) Let X be the 2nd Brocard point of pedal triangle of 1st Brocard point. Let Y be the 1st Brocard point of pedal triangle of 2nd Brocard point. Then X(3398) is the vertex conjugate of X and Y.

(2) Let U be the circle {{X(371),X(372),PU(1),PU(39)}} and V the 2nd Brocard circle. Then X(3398) is the center of the inverse-in-U of V.

(3) Let A'B'C' be the 3rd Brocard triangle. Let Oa be the circumcenter of A'BC, and define Ob and Oc cyclically. The lines AOa, BOb, COc concur in X(3398).

X(3398) lies on these lines: 3,6   5,83   140,325   384,2782   597,1576

X(3398) = midpoint of X(1342) and X(1343)
X(3398) = isogonal conjugate of X(3399)
X(3398) = inverse-in-Brocard-circle of X(3095)
X(3398) = crosssum of X(3102) ande X(3103)
X(3398) = harmonic center of circumcircle and 1st Lemoine circle
X(3398) = harmonic center of 1st and 2nd Brocard circles
X(3398) = X(3)-of-5th-anti-Brocard-triangle
X(3398) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (32,182,3), (371,372,3094), (1687,1688,39)

### X(3399) = SEC(A - 2ω) POINT

Trilinears    sec(A - 2ω) : :

X(3399) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

Let A'B'C' be the 1st Brocard triangle. Let Oa be the circumcenter of A'BC, and define Ob and Oc cyclically. The lines AOa, BOb, COc concur in X(3399). (Randy Hutson, July 20, 2016)

Let A' be the apex of the isosceles triangle BA'C constructed inward on BC such that ∠A'BC = ∠A'CB = ω. Define B' and C' cyclically. Let Oa be the circumcenter of BA'C, and define Ob and Oc cyclically. The lines AOa, BOb, COc concur in X(3399). (Randy Hutson, July 20, 2016)

Let A' be the apex of the isosceles triangle BA'C constructed outward on BC such that ∠A'BC = ∠A'CB = 2ω. Define B' and C' cyclically. Let Ha be the orthocenter of BA'C, and define Hb and Hc cyclically. The lines AHa, BHb, CHc concur in X(3399). (Randy Hutson, July 20, 2016)

X(3399) lies on the Kiepert hyperbola and this line: 5,1916   262,1506   597,1576

X(3399) = isogonal conjugate of X(3398)
X(3399) = cevapoint of X(3102) and X(3103)

### X(3400) = TAN(A - 2ω) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = tan(A - 2ω)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3400) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3400) lies on the cubic K457.

X(3400) = isogonal conjugate of X(3401)

### X(3401) = COT(A - 2ω) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cot(A - 2ω)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3401) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3401) lies on the cubic K457 and these lines: 1,1581   63,1934

X(3401) = isogonal conjugate of X(3400)

### X(3402) = TAN(A - ω) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = tan(A - ω)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3402) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3402) lies on the cubic K457 and these lines: 1,1755   42,263   1923,1973

X(3402) = isogonal conjugate of X(3403)

### X(3403) = COT(A - ω) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cot(A - ω)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3403) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3403) lies on the cubic K457 and these lines: 1,75   9,1921   19,1969   57,1920   63,561   76,1423   1707,1965

X(3403) = isogonal conjugate of X(3402)
X(3403) = isotomic conjugate of X(2186)

### X(3404) = TAN(A + ω) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = tan(A + ω)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3404) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3404) lies on the cubic K457 and these lines: 1,163   63,561   2157,2624

X(3404) = isogonal conjugate of X(3405)
X(3404) = cevapoint of X(38) and X(2236)
X(3405) = crosssum of X(1755) and X(1959)
X(3405) = crosspoint of X(1821) and X(1910)

### X(3405) = COT(A + ω) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cot(A + ω)
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = cos(A + 2πomega)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3405) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3405) lies on the cubic K457 and these lines: 1,82   19,1969   798,812

X(3405) = isogonal conjugate of X(3404)
X(3405) = cevapoint of X(1755) and X(1959)
X(3405) = crosssum of X(38) and X(2236)

### X(3406) = SEC(A + 2ω) POINT

Trilinears    sec(A + 2ω) : :

Let A' be the apex of the isosceles triangle BA'C constructed outward on BC such that

Let A' be the apex of the isosceles triangle BA'C constructed inward on BC such that ∠A'BC = ∠A'CB = 2ω. Define B' and C' cyclically. Let Ha be the orthocenter of BA'C, and define Hb and Hc cyclically. The lines AHa, BHb, CHc concur in X(3406). (Randy Hutson, July 20, 2016)

X(3406) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3406) lies on the Kiepert hyperbola and these lines: 3,1916   4,1691   76,182

X(3406) = isogonal conjugate of X(3095)

### X(3407) = CSC(A + 2ω) POINT

Trilinears    csc(A + 2ω)

Let A'B'C' be the 1st Brocard triangle. X(3407) is the radical center of the circumcircles of A'BC, B'CA, C'AB. (Randy Hutson, July 20, 2016)

Let A'B'C' be the 1st Brocard triangle and A"B"C" be the 1st anti-Brocard triangle. Let A* be the diagonal crosspoint of trapezoid B'C'C"B" (i.e., the intersection of lines B'C" and C'B"); define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(3407). (Randy Hutson, July 20, 2016)

X(3407) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3407) lies on the Kiepert hyperbola and these lines: 1,1501   6,1916   76,384   321,2205   983,985

X(3407) = isogonal conjugate of X(3094)
X(3407) = isotomic conjugate of X(3314)
X(3407) = cevapoint of X(32) and X(182)

### X(3408) = TAN(A + 2ω) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = tan(A + 2ω)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3408) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3408) lies on the cubic K457 and this line: 63,1934

X(3408) = isogonal conjugate of X(3409)

### X(3409) = COT(A + 2ω) POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cot(A + 2ω)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3409) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).

X(3409) lies on the cubic K457 and this line: 1,1917

X(3409) = isogonal conjugate of X(3408)

### X(3410) = INTERSECTION OF LINES X(2)X(98) AND X(4)X(93)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[abc + 2(b3cos B + c3cos C - a3cos A)]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3410) lies on these lines: 2,98   4,93   5,1199   23,343   68,1173   69,1369   323,427   1209,1614

### X(3411) = 1st BROCARD-KIEPERT-FERMAT CUSP

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos A - 2 sin B sin C - 3 cos(A +π/3)

Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = cos A + 4 sin B sin C - 33/2sin A

Barycentrics   h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = 16S2 - 2*33/2a2S + a2SA, where S = 2(area(ABC))

Barycentrics   h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = 20S2 - 2*33/2S - SBSC

X(3411) and X(3412) occur in Bernard Gibert's study of the Brocard-Kiepert quartic, which has three cusps: X(20) with Euler line as tangent; X(3411) with line X(5)X(13) as tangent, and X(3412) with line X(5)X(14) as tangent.

X(3411) lies on the Brocard-Kiepert quartic, Q073, and on these lines: 5,13   14,382   16,20   61,631   398,548   630,3180

### X(3412) = 2nd BROCARD-KIEPERT-FERMAT CUSP

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos A - 2 sin B sin C - 3 cos(A -π/3)

Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = cos A + 4 sin B sin C + 33/2sin A

Barycentrics   h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = 16S2 + 2*33/2a2S + a2SA

Barycentrics   h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = 20S2 + 2*33/2S - SBSC

X(3412) lies on the Brocard-Kiepert quartic, Q073, and on these lines: 5,14   13,382   15,20   62,631   397,548   629,3181

### X(3413) = 1st KIEPERT INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) =1/x(1379)

As the isogonal conjugate of a point on the circumcircle, X(3413) lies on the line at infinity.

The asymptotes of the Kiepert hyperbola meet the infinity line in X(3413) and X(3414). See the note at X(2574).

X(3413) lies on the Kiepert hyperbola and these (parallel) lines: 2,1341   30,511   98,1380   99,1379   114,2040   115,2028   1569,2029

X(3413) = isogonal conjugate of X(1379)
X(3413) = complementary conjugate of X(2039)
X(3413) = isotomic conjugate of X(6190)
X(3413) = infinite point of major axis of Steiner inellipse/circumellipse
X(3413) = bicentric difference of PU(i) for these i: 116, 118
X(3413) = ideal point of PU(i) for these i: 116, 118

### X(3414) = 2nd KIEPERT INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) =1/x(1380)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

As the isogonal conjugate of a point on the circumcircle, X(3414) lies on the line at infinity.

X(3414) lies on the Kiepert hyperbola and these (parallel) lines: 2,1340   30,511   98,1379   99,1380   114,2039   115,2029   1569,2028

X(3414) = isogonal conjugate of X(1380)
X(3414) = complementary conjugate of X(2040)
X(3414) = isotomic conjugate of X(6189)
X(3414) = infinite point of minor axis of Steiner inellipse/circumellipse
X(3414) = bicentric difference of PU(i) for these i: 117, 119
X(3414) = ideal point of PU(i) for these i: 117, 119

Vertex Conjugates
Suppose that U = u : v : w and X = x : y : z (trilinears) are points not on a sideline of ABC. Let

f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)].

The U-vertex conjugate of X is the point

f(a,b,c) : f(b,c,a) : f(c,a,b).

For a geometric interpretation, let T be the vertex triangle of the circumcevian triangles, AUBUCU and AXBXCX, of U and X; viz., the sidelines of T are AUAX, BUBX, CUCX. Then T is perspective to ABC, and the perspector is the U-vertex conjugate of X.

The definition of vertex conjugate allows X = U. To extend the geometric interpretation to the case that X = U, as X approaches U, the vertex triangle approaches a limiting triangle which we call the tangential triangle of U, a triangle perspective to ABC with perspector U-vertex conjugate of U.

The appearance of a row I, J, K in the following tables signifies that the X(i)-vertex conjugate of X(j) is X(K).

IJK
1156
123415
1384
143417
162163
173418
193420
1193422
1561
1573423
15858
2225
233424
243425
261383
2323407
2251251
252323
3364
344
363426
373427
3203346
3403345
356945
3643
3841
314903447
314983348
321313183
331823354
443
553432
666
773433
883434
991436
10103437

Among properties of vertex conjugation are these:

1. X(3)-vertex conjugation maps the Darboux cubic to the Darboux cubic. The appearance of (i,j) in the following list means that X(i) is on the Darboux cubic and that X(j) = X(3)-vertex conjugate of X(i):

(1,84), (3,64), (4,4), (20,3346), (40,3345), (1490,3347), (1498,3348), (2131,3183), (3182,3354)

2. The fixed point of U-vertex conjugation is the 1st Saragossa point of U. (Saragossa points are defined just before X(1166).) The appearance of (i,j) in the following list means that the 1st Saragossa point of X(i) is X(j):

(1,58), (2,251), (3,4), (4,54), (5,1166), (6,6), (7,3449), (8,3450), (9,3451), (21,961), (55,57)

### X(3415) = X(1)-VERTEX CONJUGATE OF X(2)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)],
where u : v : w = X(1) and x : y : z = X(2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2/[a3 - (b + c)(b2 + c2)]     (E. Danneels)

X(3415) lies on these lines: 22,55   41,386

X(3415) = isogonal conjugate of X(3416)

### X(3416) = ISOGONAL CONJUGATE OF X(3415)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/x(3415)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a3 - (b + c)(b2 + c2)     (E. Danneels)

X(3416) lies on these lines: 1,141   2,1386   6,10   7,8   40,1503   66,72   200,223   321,1836   355,511   608,1861   612,1211   613,1737   1376,1460

X(3416) = midpoint of X(8) and X(69)
X(3416) = reflection of X(i) in X(j) for these (i,j): (1,141), (6,10)
X(3416) = isogonal conjugate of X(3415)
X(3416) = anticomplement of X(1386)
X(3416) = crossdifference ofevery pair of points on the line X(834)X(2483)

### X(3417) = X(1)-VERTEX CONJUGATE OF X(4)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)],
where u : v : w = X(1) and x : y : z = X(4)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2/[a4 - a3(b + c) + 2a2bc + a(b + c)(b - c)2 - (b2 - c2)2]     (E. Danneels)

X(3417) lies on these lines: 4,2217   24,56   36,47   48,573

X(3417) = isogonal conjugate of X(355)

### X(3418) = X(1)-VERTEX CONJUGATE OF X(7)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)],
where u : v : w = X(1) and x : y : z = X(7)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2/[a4 - a3(b + c) + a(b + c)(b2 + c2) - (b2 - c2)2]     (E. Danneels)

X(3418) lies on this line: 36,48

X(3418) = isogonal conjugate of X(3419)

### X(3419) = ISOGONAL CONJUGATE OF X(3418)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/x(3418)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a4 - a3(b + c) + a(b + c)(b2 + c2) - (b2 - c2)2     (E. Danneels)

X(3419) lies on these lines: 1,442   4,8   5,78   9,80   11,997   30,63   40,1726   100,1006   150,2893   226,519   381,908   392,497   405,950   443,938   474,1210   960,1479   1060,2000   1104,1714   1376,1737   2475,2894

X(3419) = midpoint of X(8) and X(3434)
X(3419) = reflection of X(i) in X(j) for these (i,j): (1,2886), (55,10)
X(3419) = isogonal conjugate of X(3418)
X(3419) = inverse-in-Fuhrmann circle of X(72)

### X(3420) = X(1)-VERTEX CONJUGATE OF X(9)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)],
where u : v : w = X(1) and x : y : z = X(9)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2/[a4 - 4abc(b + c - a) - (b2 - c2)2]     (E. Danneels)

X(3420) lies on these lines: 3,392   25,104   859,1444   1450,2122   1811,2932   2163,2291

X(3420) = isogonal conjugate of X(3421)

### X(3421) = ISOGONAL CONJUGATE OF X(3420)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/x(3420)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a4 - 4abc(b + c - a) - (b2 - c2)2     (E. Danneels)

X(3421) lies on these lines: 1,2551   2,495   4,8   40,2123   57,388   100,376   144,153   145,1058   150,668   200,515   219,1249   497,519   527,1478   529,1376   631,2975

X(3421) = midpoint of X(8) and X(329)
X(3421) = reflection of X(i) in X(j) for these (i,j): (57,10), (2096,3359)
X(3421) = isogonal conjugate of X(3420)
X(3421) = anticomplement of X(999)

### X(3422) = X(1)-VERTEX CONJUGATE OF X(19)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)],
where u : v : w = X(1) and x : y : z = X(19)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2/[a4 + 2a2bc - (b2 - c2)2]     (E. Danneels)

X(3422) lies on these lines: 1,24   3,47   29,1479   35,78   36,77   55,1807   219,2174   573,2359   1065,1478   2163,2291   2301,2338

X(3422) = isogonal conjugate of X(1478)
X(3422) =X(1064)-cross conjugate of X(1)

### X(3423) = X(1)-VERTEX CONJUGATE OF X(57)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)],
where u : v : w = X(1) and x : y : z = X(57)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2/[a3 - a2(b + c) + a(b + c)2 - (b + c)(b - c)2]     (E. Danneels)

X(3423) lies on these lines: 3,41   7,105   25,57   31,222   55,63   513,884   1037,1362   1790,2194   2163,2291

X(3423) = isogonal conjugate of X(2550)

### X(3424) = X(2)-VERTEX CONJUGATE OF X(3)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)],
where u : v : w = X(2) and x : y : z = X(3)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 1/[a4 + 2a2(b2 + c2) - 3b4 - 3c4 - 2b2c2]     (E. Danneels)

X(3424) lies on these lines: 2,154   20,76   193,1916   253,1297   385,2996   671,2794   1499,2394

X(3424) = isogonal conjugate of X(1350)
X(3424) = perspector of ABC and anticomplementary triangle of Artzt triangle

### X(3425) = X(2)-VERTEX CONJUGATE OF X(4)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)],
where u : v : w = X(2) and x : y : z = X(4)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2/[a6 - a4(b2 + c2) + a2(b2 + c2)2 - (b2 + c2)(b2 - c2)2]     (E. Danneels)

X(3425) lies on these lines: 2,2351   3,315   4,2353   22,184   32,232   98,264   378,2794   523,878

X(3425) = isogonal conjugate of X(1352)

### X(3426) = X(3)-VERTEX CONJUGATE OF X(6)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)],
where u : v : w = X(3) and x : y : z = X(6)
Trilinears        1/(3 cos A - sin B sin C) : 1/(3 cos B - sin C sin A) : 1/(3 cos C - sin A sin B)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2/[5a4 - 4a2(b2 + c2) - (b2 - c2)2]     (E. Danneels)

X(3426) lies on these lines: 3,1495   6,1597   25,74   30,69   54,1593   64,1598   67,2777   68,382   73,1480   248,1384   265,541   381,1514   879,1499   895,1351

X(3426) = isogonal conjugate of X(376)

### X(3427) = X(3)-VERTEX CONJUGATE OF X(7)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)],
where u : v : w = X(3) and x : y : z = X(7)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b),
where g(a,b,c) = 1/[a5 - a4(b + c) - 2a3(b2 + c2) + 2a2(b + c)(b - c)2 + a(b4 + 6b2c2 + c4) - (b + c)(b - c)4]     (E. Danneels)

X(3427) lies on these lines: 9,515   104,1617   943,944   1156,2829

X(3427) = isogonal conjugate of X(3428)

### X(3428) = ISOGONAL CONJUGATE OF X(3427)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/x(3427)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b),
where g(a,b,c) = a2[a5 - a4(b + c) - 2a3(b2 + c2) + 2a2(b + c)(b - c)2 + a(b4 + 6b2c2 + c4) - (b + c)(b - c)4]     (E. Danneels)

X(3428) lies on these lines: 1,3   4,958   6,1064   8,411   20,2894   21,962   101,102   104,376   212,1457   221,255   347,934   405,946   516,993   602,1191   859,2328   945,1794   953,2742   1001,1006   1042,1496   1808,2716   1753,1824   2178,2256

X(3428) = midpoint of X(20) and X(3434)
X(3428) = reflection of X(i) in X(j) for these (i,j): (4,2886), (55,3), (1012,993)
X(3428) = isogonal conjugate of X(3427)

### X(3429) = X(3)-VERTEX CONJUGATE OF X(10)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)],
where u : v : w = X(3) and x : y : z = X(10)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b),
nbsp;        where g(a,b,c) = 1/[a5 + a3bc + a2(b + c)(2b2 - bc + 2c2) - a(b + c)2(b2 - bc + c2) - (b + c)(2b4 - b3c + 2b2c2 - bc3 + 2c4)]     (E. Danneels)

X(3429) lies on this line: 10,1503

X(3429) = isogonal conjugate of X(3430)

### X(3430) = ISOGONAL CONJUGATE OF X(3429)

Trilinears    1/x(3429) : :\
Barycentrics    2[a5 + a3bc + a2(b + c)(2b2 - bc + 2c2) - a(b + c)2(b2 - bc + c2) - (b + c)(2b4 - b3c + 2b2c2 - bc3 + 2c4)] : :      (E. Danneels)

X(3430) lies on these lines: 3,6   20,1330   72,1782   78,1763   101,1297   169,936   404,1730   842,2705   951,1427   1293,2842   1490,1766   2360,2915   2702,2710

X(3430) = midpoint of X(20) and X(1330)
X(3430) = reflection of X(58) in X(3))
X(3430) = isogonal conjugate of X(3429)

### X(3431) = X(4)-VERTEX CONJUGATE OF X(6)

Trilinears    a/[a2vwyz - ux(bw + cv)(bz + cy)], where u : v : w = X(4) and x : y : z = X(6)
Barycentrics    a2/[a4 + a2(b2 + c2) - 2(b2 - c2)2] : :     (E. Danneels)
Barycentrics    (SB + SC)/(S^2 + 3 SB SC) : :

Let A'B'C' be the orthocentroidal triangle. X(3431) is the radical center of the circumcircles of A'BC, B'CA, C'AB. (Randy Hutson, June 27, 2018)

X(3431) lies on these lines: 2,265   3,323   4,1495   6,186   68,631   248,574

X(3431) = isogonal conjugate of X(381)
X(3431) = X(566)-cross conjugate of X(2)
X(3431) = trilinear pole of line X(526)X(647)
X(3431) = perspector of circle (X(3),R/2)
X(3431) = perspector of ABC and complement of Ehrmann vertex-triangle

### X(3432) = X(5)-VERTEX CONJUGATE OF X(5)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)],
where u : v : w = X(5) and x : y : z = X(5)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c)
= a2/[a10 - 3a8(b2 + c2) + a6(4b4 + 5b2c2 + 4c4) - 2a4(b2 + c2)(2b4 - b2c2 + 2c4) + 3a2(b2 - c2)2(b4 + b2c2 + c4) - (b2 + c2)(b2 - c2)4]     (E. Danneels)

X(3432) lies on these lines: 5,1601   49,52

X(3432) = isogonal conjugate of X(2888)
X(3432) = crosssum of X(195) and X(2917)

### X(3433) = X(7)-VERTEX CONJUGATE OF X(7)

Trilinears    a/[a2vwyz - ux(bw + cv)(bz + cy)] : : , where u : v : w = X(7) and x : y : z = X(7)
Barycentrics   a2/[a3 - a2b - a2c + ab2 + ac2 - (b + c)(b - c)2] : :      (E. Danneels)

Let A'B'C' be the intouch triangle. Let A" be the crosspoint of the circumcircle intercepts of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(3433). (Randy Hutson, July 31 2018)

X(3433) lies on these lines: 3,518   7,1486   48,672   56,1876   104,378   603,1458   692,1037   1437,1780

X(3433) = isogonal conjugate of X(3434)
X(3433) = X(2175)-cross conjugate of X(6)

### X(3434) = ISOGONAL CONJUGATE OF X(3433)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/x(3433)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a3 - a2b - a2c + ab2 + ac2 - (b + c)(b - c)2      (E. Danneels)

X(3434) lies on these lines: 1,224   2,11   4,8   10,1479   20,2894   30,956   63,516   69,674   75,1370   78,946   145,388   152,2807   200,908   226,2900   443,1058   474,496   518,1836   519,1478   595,1714   614,1738   1004,1617   1484,2932

X(3434) = midpoint of X(8) and X(3419)
X(3434) = reflection of X(i) in X(j) for these (i,j): (8,3419), (20,3428), (55,2885), (145,2099)
X(3434) = isogonal conjugate of X(3433)
X(3434) = inverse-in-Fuhmann circle of X(3436)
X(3434) = anticomplement of X(55)
X(3434) = anticomplementary conjuate of X(144)
X(3434) = homothetic center of anticomplementary triangle and cross-triangle of ABC and inner Johnson triangle
X(3434) = homothetic center of inner Johnson triangle and cross-triangle of ABC and inner Johnson triangle

### X(3435) = X(8)-VERTEX CONJUGATE OF X(8)

Trilinears    a/[a2vwyz - ux(bw + cv)(bz + cy)] : : , where u : v : w = X(8) and x : y : z = X(8)
Barycentrics    a2/[a4 + 2a2bc - 2abc(b + c) - (b2 - c2)2] : :    (E. Danneels)

Let A'B'C' be the extouch triangle. Let A" be the crosspoint of the circumcircle intercepts of line B'C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(3435). (Randy Hutson, July 31 2018)

X(3435) lies on these lines: 3,960   8,197   24,104   25,2217   48,836   56,1452   603,1193

X(3435) = isogonal conjugate of X(3436)
X(3435) = X(i)-cross conjugate of X(j) for these (i,j): (1364,513), (1397,6)

### X(3436) = ISOGONAL CONJUGATE OF X(3435)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/x(3445)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a4 + 2a2bc - 2abc(b + c) - (b2 - c2)2     (E. Danneels)

X(3436) lies on these lines: 1,908   2,12   4,8   5,956   10,46   20,100   69,313   78,515   144,1654   145,497   271,1542   315,668   341,1370   405,495   452,1621   498,993   518,1837   519,1479

X(3436) = reflection of X(i) in X(j) for these (i,j): (46,10), (56,1329), (145,2098)
X(3436) = isogonal conjugate of X(3435)
X(3436) = inverse-in-Fuhmann circle of X(3434)
X(3436) = anticomplement of X(56)
X(3436) = anticomplementary conjuate of X(145) X(3436) = X(478)-cross conjugate of X(2)
X(3436) = isotomic conjugate of X(8048)
X(3436) = homothetic center of anticomplementary triangle and cross-triangle of ABC and outer Johnson triangle
X(3436) = homothetic center of outer Johnson triangle and cross-triangle of ABC and outer Johnson triangle

### X(3437) = X(10)-VERTEX CONJUGATE OF X(10)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)],
where u : v : w = X(10) and x : y : z = X(10)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2/[a4 + a3(b + c) + a2bc - a(b + c)(b2 + c2) - (b + c)2(b2 - bc + c2)]     (E. Danneels)

X(3437) lies on these lines: 10,199   35,228   184,386   2174,2200

X(3437) = isogonal conjugate of X(1330)
X(3437) = X(2206)-cross conjugate of X(6)
X(3437) = crosssum of X(1654) and X(3151)

### X(3438) = X(13)-VERTEX CONJUGATE OF X(13)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)],
where u : v : w = X(13) and x : y : z = X(13)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3438) lies on these lines: 3,299   13,1605   2174,2200

X(3438) = isogonal conjugate of X(621)
X(3438) = X(186)-cross conjugate of X(3439)
X(3438) = crosssum of X(616) and X(628)

### X(3439) = X(14)-VERTEX CONJUGATE OF X(14)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)],
where u : v : w = X(14) and x : y : z = X(14)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3439) lies on these lines: 3,298   14,1606   15,184

X(3439) = isogonal conjugate of X(622)
X(3439) = inverse-in-circumcircle of X(3479)
X(3439) = X(186)-cross conjugate of X(3438)
X(3439) = crosssum of X(616) and X(627)

### X(3440) = X(15)-VERTEX CONJUGATE OF X(15)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)],
where u : v : w = X(15) and x : y : z = X(15)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3440) lies on the Neuberg cubic and these lines: 15,1495   30,298   399,1338   2132,3441

X(3440) = isogonal conjugate of X(616)
X(3440) = X(74)-cross conjugate of X(3441)

### X(3441) = X(16)-VERTEX CONJUGATE OF X(16)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)],
where u : v : w = X(16) and x : y : z = X(16)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3441) lies on the Neuberg cubic and these lines: 16,1495   30,299   399,1337   2132,3440

X(3441) = isogonal conjugate of X(617)
X(3441) = X(74)-cross conjugate of X(3440)

### X(3442) = X(17)-VERTEX CONJUGATE OF X(17)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)],
where u : v : w = X(17) and x : y : z = X(17)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3442) lies on these lines: 3,303   17,1607   62,184

X(3442) = isogonal conjugate of X(633)

### X(3443) = X(18)-VERTEX CONJUGATE OF X(18)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)],
where u : v : w = X(18) and x : y : z = X(18)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3443) lies on these lines: 3,302   18,1608   61,184

X(3443) = isogonal conjugate of X(634)

### X(3444) = X(37)-VERTEX CONJUGATE OF X(37)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)],
where u : v : w = X(37) and x : y : z = X(37)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2/[a3 + a2(b + c) - a(b2 + bc + c2) - (b + c)(b2 + c2)]     (E. Danneels)

X(3444) lies on these lines: 2,1029   35,37   42,2174   759,1989   1399,1400

X(3444) = isogonal conjugate of X(2895)
X(3444) = cevapoint of X(667) and X(3124)
X(3444) = X(1333)-cross conjugate of X(6)

### X(3445) = X(56)-VERTEX CONJUGATE OF X(56)

Trilinears        a/(3a - b - c) : b/(3b - c - a) : c/(3c - a - b)
Barycentrics   a2/(3a - b - c) : b2/(3b - c - a) : c2/(3c - a - b)

Let A'B'C' be the tangential triangle of ABC, and let L be the line through X(1) parallel to BC. Let A'' = L∩B'C', and define B'' and C'' cyclically. Let A* = B'B''∩C'C'', and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(3445); also, the lines A'A*, B'B*, C'C* concur in X(3052). (Angel Montesdeoca, April 29, 2016)

X(3445) = perspector of ABC and the inner-mixtilinear tangetns triangle (see X(11051).

X(3445) lies on these lines: 1,474   2,1222   3,106   6,1201   8,1120   34,1319   56,1149   58,999   87,1001   220,1015   244,2098   269,1279   937,1104   958,979   995,1126   996,1125   1220,2899   1357,2308   1388,1411   1413,1457   2176,2279   2191,2646   2256,2983

X(3445) = isogonal conjugate of X(145)
X(3445) = anticomplement of X(2885)
X(3445) = cevapoint of X(663) and X(1015)
X(3445) = X(i)-cross conjugate of X(j) for these (i,j): (55,6), (1357,513), (2347,57)
X(3445) = crosspoint of X(1) and X(2137)
X(3445) = crosssum of X(i) and X(j) for these (i,j): (1,2136), (1743,3158)
X(3445) = trilinear pole of line X(649)X(6363)
X(3445) = perspector of ABC and unary cofactor of triangle T(-2,1)
X(3445) = crossdifference of every pair of points on line X(2976)X(3667) (the radical axis of incircle and AC-incircle)

### X(3446) = X(513)-VERTEX CONJUGATE OF X(513)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)],
where u : v : w = X(513) and x : y : z = X(513)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2/[a3 - a2b - a2c + a(b2 - bc + c2) - (b + c)(b - c)2]      (E. Danneels)

X(3446) lies on these lines: 36,518   840,1618

X(3446) = isogonal conjugate of X(149)
X(3446) = X(692)-cross conjugate of X(6)

### X(3447) = X(523)-VERTEX CONJUGATE OF X(523)

Trilinears    a/[a2vwyz - ux(bw + cv)(bz + cy)] : : , where u : v : w = x : y : z = X(523)
Trilinears    a/[a6 - a4(b2 + c2) + a2(b4 - b2c2 + c4) - (b2 + c2)(b2 - c2)2]      (E. Danneels)

Let E be the Euler line and TATBTC the tangential triangle of ABC. Let DA = E∩TBTC, and define DB and DC cyclically. Let A' = DBTB∩DCTC, and define B' and C' cyclically. Then A'B'C' is perspective to ABC, and the perspector is X(3447). For a sketch, click X(3447)andX(7669). (Angel Montesdeoca, April 22, 2016)

X(3447) lies on these lines: 23,325   50,232   264,2453   511,2070   2079,2485

X(3447) = isogonal conjugate of X(3448)
X(3447) = X(1576)-cross conjugate of X(6)
X(3447) = perspector of ABC and reflection of tangential triangle in Euler line

### X(3448) = ISOGONAL CONJUGATE OF X(3447)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/x(3447)
Trilinears        g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = bc[a6 - a4(b2 + c2) + a2(b4 - b2c2 + c4) - (b2 + c2)(b2 - c2)2]      (E. Danneels)

Let A'B'C' be the orthocentroidal triangle and A"B"C" the anti-orthocentroidal triangle. Let A* be the reflection of A" in B'C', and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(74), and X(3448) = centroid of A*B*C*. (Randy Hutson, December 10, 2016)

X(3448) lies on these lines: 2,98   3,2888   4,94   5,399   10,2948   20,68   23,1503   66,193   67,69   141,2930   148,690   150,2774   151,2779   152,2772   153,355   246,2782   323,858   427,1353   631,1511   1330,2842   1853,1993   2836,2895   2918,2931

X(3448) = reflection of X(i) in X(j) for these (i,j):
(4,265), (20,74), (69,67), (110,125), (146,4), (193,895), (323,858), (399,5), (2892,66), (2930,141), (2948,10)

X(3448) = isogonal conjugate of X(3447)
X(3448) = inverse-in-Fuhrmann circle of X(2475)
X(3448) = anticomplement of X(110)
X(3448) = anticomplementary conjugate of X(523)
X(3448) = X(850)-Ceva conjugate of X(2)
X(3448) = inverse-in-polar-circle of X(1112)
X(3448) = center of rectangular hyperbola through X(4), X(8), and the extraversions of X(8)
X(3448) = X(3060)-of-anti-orthocentroidal-triangle

### X(3449) = 1st SARAGOSSA POINT OF X(7)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[x(bz + cy)], where x : y : z = X(7)
Trilinears        g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a/[a(b2 + c2) - (b + c)(b - c)2]      (E. Danneels)
Barycentrics   ag(a,b,c) : bg(b,c,a) : cg(c,a,b)

Saragossa points are discussed just before X(1166).

X(3449) lies on these lines: 6,1602   7,2175   518,2330   572,672

X(3449) = isogonal conjugate of X(2886)
X(3449) = cevapoint of X(6) and X(2175)

### X(3450) = 1st SARAGOSSA POINT OF X(8)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(x(bz + cy), where x : y : z = X(8)
Trilinears        g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a/{(b + c - a)[a(b2 + c2) + (b + c)(b - c)2]}      (E. Danneels)
Barycentrics   ag(a,b,c) : bg(b,c,a) : cg(c,a,b)

X(3450) lies on these lines: 6,1603   8,1397   35,572   960,1319   1193,2003

X(3450) = isogonal conjugate of X(1329)
X(3450) = complementary conjugate of X(3452)
X(3450) = cevapoint of X(6) and X(1397)
X(3450) = X(6)-crossconjugate of X(2985)

### X(3451) = 1st SARAGOSSA POINT OF X(9)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(x(bz + cy), where x : y : z = X(9)
Trilinears        g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a/{(b + c - a)[a(b + c) + (b - c)2]}      (E. Danneels)
Barycentrics   ag(a,b,c) : bg(b,c,a) : cg(c,a,b)

X(3451) lies on these lines: 6,1604   9,604   55,572   284,1404   333,1412   1108,2161   1400,2316   2259,2317

X(3451) = isogonal conjugate of X(3452)
X(3451) = cevapoint of X(i) and X(j) for these (i,j): (6,604), (41,3052)
X(3451) = X(667)-cross conjugate of X(109)

### X(3452) = ISOGONAL CONJUGATE OF X(3451)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/(x(3451)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b + c - a)[ab + ac + (b - c)2]      (E. Danneels)

X(3452) lies on these lines: 1,2551   2,7   4,936   5,10   11,210   69,1997   72,1210   78,950   118,123   120,124   200,497   281,1848   312,2321   333,645   345,2325   515,997   516,1376   631,2096   958,999   962,1706   1656,2095   1698,2093   1699,2550   1764,2183   2324,2999

X(3452) = midpoint of X(i) and X(j) for these (i,j): (1,3421), (57,329), (200,497)
X(3452) = reflection of X(999) in X(1125)
X(3452) = isogonal conjugate of X(3451)
X(3452) = complement of X(57)
X(3452) = complementary conjugate of X(3450)
X(3452) = X(668)-Ceva conugate of X(522)
X(3452) = crosspoint of X(2) and X(312)
X(3452) = crosssum of X(i) and X(j) for these (i,j): (6,604), (41,3052)

### X(3453) = 1st SARAGOSSA POINT OF X(10)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(x(bz + cy), where x : y : z = X(10)
Trilinears        g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a/{(b + c)[a(b2 + c2) + b3 + c3]}      (E. Danneels)
Barycentrics   ag(a,b,c) : bg(b,c,a) : cg(c,a,b)

X(3453) lies on these lines: 60,386   261,1078   404,849

X(3453) = isogonal conjugate of X(3454)
X(3453) = cevapoint of X(i) and X(j) for these (i,j): (6,2206), (2194,2220)
X(3453) = X(1324)-cross conjugate of X(759)

### X(3454) = ISOGONAL CONJUGATE OF X(3453)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/(x(3453)
Trilinears        g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = bc(b + c)[a(b2 + c2) + b3 + c3]      (E. Danneels)

X(3454) lies on these lines: 2,58   5,141   10,12   34,860   44,1213   118,127   121,125   306,2901   429,1828   519,1834   1046,1698   1089,1230   1104,1125   1228,1930

X(3454) = midpoint of X(i) and X(j) for these (i,j): (4,3430), (58,1330)
X(3454) = isogonal conjugate of X(3453)
X(3454) = complement of X(58)
X(3454) = complementary conjugate of X(1125)
X(3454) = crosspoint of X(2) and X(313)
X(3454) = crosssum of X(i) and X(j) for these (i,j): (6,2206), (2194,2220)

### X(3455) = X(13)-VERTEX CONJUGATE OF X(14)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)],
where u : v : w = X(13) and x : y : z = X(14)
Barycentrics    a^2/(a^4 - b^4 - c^4 + b^2 c^2) : :

X(3455) lies on these lines: 3,67   22,543   23,671   25,115   39,1576   98,186   99,1799   184,574   187,2393   228,2157   378,2794   1976,2088

X(3455) = isogonal conjugate of X(316)
X(3455) = inverse-in-circumcircle of X(67)
X(3455) = cevapoint of X(39) and X(187)
X(3455) = crosspoint of X(111) and X(1177)
X(3455) = crosssum of X(524) and X(858)
X(3455) = vertex conjugate of PU(40)
X(3455) = X(75)-isoconjugate of X(23)

### X(3456) = X(17)-VERTEX CONJUGATE OF X(18)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)],
where u : v : w = X(17) and x : y : z = X(18)
Barycentrics    a^2/(a^4 - b^4 - c^4 - b^2 c^2) : :

X(3456) lies on these lines: 3,2916   23,1799

X(3456) = isogonal conjugate of X(7768)
X(3456) = crosssum of X(69) and X(1369)

### X(3457) = X(13)-VERTEX CONJUGATE OF X(15)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)],
where u : v : w = X(13) and x : y : z = X(15)
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = (sin2A)(sin A/2) sec(A - π/6)

Let A' be the free vertex of the equilateral triangle constructed outwardly on BC, and define B' and C' cyclically. X(3457) is the barycentric product A'*B'*C'. (Randy Hutson, July 31 2018)

X(3457) lies on these lines: 2,13   15,1337   32,3124   37,1250   62,110   300,308

X(3457) = isogonal conjugate of X(298)
X(3457) = X(1495)-cross conjugate of X(3458)
X(3457) = crosspoint of X(6) and X(3440)
X(3457) = crosssum of X(2) and X(616)
X(3457) = X(15)-isoconjugate of X(75)
X(3457) = barycentric product of circumcircle intercepts of inner Napoleon circle

### X(3458) = X(14)-VERTEX CONJUGATE OF X(16)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)],
where u : v : w = X(14) and x : y : z = X(16)

Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = (sin2A)(sin A/2) sec(A + π/6)

Let A' be the free vertex of the equilateral triangle constructed inwardly on BC, and define B' and C' cyclically. X(3458) is the barycentric product A'*B'*C'. (Randy Hutson, July 31 2018)

X(3458) lies on these lines: 2,14   16,1338   32,3124   37,1254  61,110   301,308

X(3458) = isogonal conjugate of X(299)
X(3458) = X(1495)-cross conjugate of X(3457)
X(3458) = crosspoint of X(6) and X(3441)
X(3458) = crosssum of X(2) and X(617)

X(3458) = X(16)-isoconjugate of X(75)
X(3458) = barycentric product of circumcircle intercepts of outer Napoleon circle

### X(3459) = ISOGONAL CONJUGATE OF X(195)

Trilinears          f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = y(195)z(195)

Let A' be the reflection of X(5) in BC, and define B' and C' cyclically. Let Oa be the circumcenter of A'BC, and define Ob and Oc cyclically. The lines AOa, BOb, COc concur in X(3459). (Randy Hutson, December 2, 2017)

X(3459) lies on the Napoleon cubic and these lines: 3,1263   4,1157   5,195   53,1601   137,252   288,1487   3336,3460   3462,3470

X(3459) = isogonal conjugate of X(195)
X(3459) = Cundy-Parry Phi transform of X(1263)
X(3459) = Cundy-Parry Psi transform of X(1157)
X(3459) = X(i)-cross conjugate of X(j) for these (i,j): (54,4), (2963,2)

### X(3460) = X(5)-CEVA CONJUGATE OF X(1)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = x(- x/u + y/v + z/w), where u : v : w = X(5), x : y : z = X(1)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3460) lies on the Napoleon cubic and these lines: 1,54   3,3469   4,484   5,2595   655,1087   1726,1745   3336,3459

X(3460) = isogonal conjugate of X(3461)
X(3460) = X(5)-Ceva conjugate of X(1)

### X(3461) = ISOGONAL CONJUGATE OF X(3460)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/x(3460)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3461) lies on the Napoleon cubic and these lines: 1,2120   5,2595   195,3468

X(3461) = isogonal conjugate of X(3460)
X(3461) = X(54)-cross conjugate of X(1)

### X(3462) = X(5)-CEVA CONJUGATE OF X(4)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = x(- x/u + y/v + z/w), where u : v : w = X(5), x : y : z = X(4)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3462) lies on the Napoleon cubic and these lines: 1,3469   2,1075   4,54   5,3463   195,2121   233,1249   499,1148   3336,3461

X(3462) = isogonal conjugate of X(3463)
X(3462) = X(5)-Ceva conjugate of X(4)

### X(3463) = ISOGONAL CONJUGATE OF X(3462)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/x(3462)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3463) lies on the Napoleon cubic and these lines: 3,2120   5,3462   3460,3468

X(3463) = isogonal conjugate of X(3462)
X(3463) = X(54)-cross conjugate of X(3)

### X(3464) = X(30)-CEVA CONJUGATE OF X(1)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = x(- x/u + y/v + z/w), where u : v : w = X(30), x : y : z = X(1)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3464) lies on the Neuberg cubic and these lines: 1,74   3,3466   4,1768   13,1652   14,1653   57,1354   1054,1724

X(3464) = X(30)-Ceva conjugate of X(1)

### X(3465) = X(30)-CEVA CONJUGATE OF X(484)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = x(- x/u + y/v + z/w), where u : v : w = X(5), x : y : z = X(484)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3465) lies on the Neuberg cubic and these lines: 1,4   30,1807   35,228   36,1736   74,484   108,2732   912,1936   1047,1048   1157,3065   1725,1758   2250,2341   3481,3483

X(3465) = reflection of X(484) in X(2222)
X(3465) = isogonal conjugate of X(3466)
X(3465) = X(i)-Ceva conjugate of X(j) for these (i,j): (30,484), (1807,1)
X(3465) = crosssum of X(654) and X(2638)
X(3465) = crossdifference of every pair of points on the line X(652)X(2260)

### X(3466) = ISOGONAL CONJUGATE OF X(3465)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/x(3465)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3466) lies on the Neuberg cubic and these lines: 1,2816   3,3464   30,1807   78,1330   109,1794   219,1761   284,1845   399,3483   484,3484   947,2817   1718,2636   2132,3065

X(3466) = isogonal conjugate of X(3465)
X(3466) = cevapoint of X(654) and X(2638)
X(3466) = X(i)-cross conjugate of X(j) for these (i,j): (74,3065), (1870,1)

### X(3467) = ISOGONAL CONJUGATE OF X(3336)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/x(3336)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3467) lies on the Napoleon cubic and these lines: 1,195   4,3460   5,79   7,499   62,1251   2120,3469   3468,3470

X(3467) = isogonal conjugate of X(3336)
X(3467) = X(i)-cross conjugate of X(j) for these (i,j): (35,1), (54,3469)

### X(3468) = X(5)-CEVA CONJUGATE OF X(3336)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = x(- x/u + y/v + z/w), where u : v : w = X(5), x : y : z = X(3336)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3468) lies on the Napoleon cubic and these lines: 1,4   5,3469   36,1410   47,1758   54,3336   62,1652   62,1653   195,3461   610,2317   1047,1718   1051,2939   3460,3463   3467,3470

X(3468) = isogonal conjugate of X(3469)
X(3468) = X(5)-Ceva conjugate of X(3336)

### X(3469) = ISOGONAL CONJUGATE OF X(3468)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/x(3468)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3469) lies on the Napoleon cubic and these lines: 1,3462   3,3460   5,3468   77,1745   2120,3467

X(3469) = isogonal conjugate of X(3468)
X(3469) = X(54)-cross conjugate of X(3467)

### X(3470) = TRILINEAR PRODUCT X(74)*X(1749)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = x(74)x(1749)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3470) lies on the Napoleon cubic and these lines: 3,74   5,1117   1138,2132   3459,3462   3467,3468

X(3470) = isogonal conjugate of X(3471)
X(3470) = X(2914)-cross conjugate of X(1157)

### X(3471) = ISOGONAL CONJUGATE OF X(3470)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/x(3471)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3471) lies on the Napoleon cubic and these lines: 3,1138   5,1117   140,523   477,550

X(3471) = isogonal conjugate of X(3470)
X(3471) = X(1511)-cross conjugate of X(30)

### X(3472) = X(20)-CEVA CONJUGATE OF X(3345)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = x(- x/u + y/v + z/w), where u : v : w = X(20), x : y : z = X(3345)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3472) lies on the Darboux cubic and these lines: 1,3348   3,3354   4,3347   20,3473   40,2131   64,3345   84,3346

X(3472) = reflection of X(3354) in X(3)
X(3472) = isogonal conjugate of X(3473)
X(3472) = X(20)-Ceva conjugate of X(3345)

### X(3473) = ISOGONAL CONJUGATE OF X(3472)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/x(3472)

X(3473) lies on the Darboux cubic and these lines: 20,3472   40,3355   1490,2130   1498,3353   3182,3183

X(3473) = isogonal conjugate of X(3472)
X(3473) = X(64)-cross conjugate of X(1490)

### X(3474) = INTERSECTION OF LINES X(4)X(46) AND X(7)X(55)

Trilinears    - cos A + cos B + cos C + cos B cos C : :
Barycentrics    3 a^3 - a^2 (b + c) - a (b - c)^2 - (b - c)^2 (b + c) : :
Barycentrics    2 a^2 (b + c - a) - (a + b + c) (a - b + c) (a + b - c) : :

X(3474) lies on these lines: 1,376   2,1155   4,46   7,55   8,529   20,65   25,1633   33,1721   40,388   42,3000   43,2635   56,962   57,497   63,2550   73,1044   79,498   109,278   144,210   165,226   196,243   200,527   212,948   329,1376   354,390   412,1118   484,1478   515,2093   528,2094   580,1777   653,1857   658,2898   1040,2263   1707,1738   1761,2345

X(3474) = reflection of X(i) in X(j) for these (i,j): (329,1376), (497,57)
X(3474) = cevapoint of X(46) and X(1721)
X(3474) = crosssum of X(652) and X(3022)
X(3474) = {X(7),X(55)}-harmonic conjugate of X(3475)

### X(3475) = INTERSECTION OF LINES X(1)X(4) AND X(7)X(55)

Trilinears    2 + cos A + cos B + cos C + cos B cos C : :
Trilinears    r + 3R + R cos B cos C : :

X(3475) lies on these lines: 1,4   2,210   7,55   12,938   142,200   165,553   329,1001   942,1788   954,1617

X(3475) = isogonal conjugate of X(3477)
X(3475) = {X(7),X(55)}-harmonic conjugate of X(3474)

### X(3476) = INTERSECTION OF LINES X(1)X(4) AND X(8)X(56)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 2 - cos A - cos B - cos C + cos B cos C
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3476) lies on these lines: 1,4   2,1319   7,528   8,56   10,1420   12,1388   43,1450   57,519   100,1470   329,529   381,1387   604,2345   952,999   956,1617   993,2078

X(3476) = reflection of X(i) in X(j) for these (i,j): (8,1376), (497,1)
X(3476) = isogonal conjugate of X(3478)

### X(3477) = ISOGONAL CONJUGATE OF X(3475)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(2 + cos A + cos B + cos C + cos B cos C)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3477) lies on these lines: 1,1827   3,1471   77,354   78,1001   219,2280

X(3477) = isogonal conjugate of X(3475)

### X(3478) = ISOGONAL CONJUGATE OF X(3476)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(2 - cos A - cos B - cos C + cos B cos C)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3478) lies on these lines: 1,1828   3,902   77,1122   219,2347   1037,1457   1473,1795

X(3478) = isogonal conjugate of X(3476)

### X(3479) = ISOGONAL CONJUGATE OF X(1337)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/x(1337)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3479) is the tangential of X(1276) on the Neuberg cubic.

X(3479) lies on the Neuberg cubic and these lines: 3,298   30,1337   3480,3484

X(3479) = isogonal conjugate of X(1337)
X(3479) = inverse-in-circumcircle of X(3439)

### X(3480) = ISOGONAL CONJUGATE OF X(1338)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/x(1338)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3480) is the tangential of X(1277) on the Neuberg cubic.

X(3480) lies on the Neuberg cubic and these lines: 3,299   30,1338   3479,3484

X(3480) = isogonal conjugate of X(1338)

### X(3481) = INTERSECTION OF LINES X(3)X(2120) and X(4)X(2121)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = P + Q + (16Px + P2 + 4PQ + 3Q2 - 24P)/(8x2 - 8x + P + 7Q),
where P = 16S2/(a + b + c)4,     Q = -1 + 4(bc + ca + ab)/(a + b + c)2,      S = 2(area of ABC);
and x = (-a + b + c)/(a + b + c)         (M. Iliev, 10/27/08)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3481) lies on the Neuberg cubic and these lines: 3,2120   4,2121   30,3482   3465,3483

X(3481) = isogonal conjugate of X(3482)

### X(3482) = ISOGONAL CONJUGATE OF X(3481)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/x(3481)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3482) lies on the Neuberg cubic and these lines: 3,2121   4,1157   30,3481

X(3482) = isogonal conjugate of X(3481)

### X(3483) = INTERSECTION OF LINES X(1)X(1157) and X(4)X(484)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3 - (b + c)(b2 + c2) + bc(b + c)P/[(b + c)U + aV],
where P = (b + c)[2a2 - (b - c)2] - a(b2 + c2),     U = a2 - (b - c)2,
and V = a2 - b2 - c2 + bc         (M. Iliev, 10/27/08)

Barycentrics   (a (a^6+a^5 b-a^4 b^2-2 a^3 b^3-a^2 b^4+a b^5+b^6+a^5 c+a^4 b c-a b^4 c-b^5 c-a^4 c^2+2 a^2 b^2 c^2-b^4 c^2-2 a^3 c^3+2 b^3 c^3-a^2 c^4-a b c^4-b^2 c^4+a c^5-b c^5+c^6))/(-a^3-a^2 b+a b^2+b^3-a^2 c-a b c-b^2 c+a c^2-b c^2+c^3) : :

X(3483) lies on the Neuberg cubic and these lines: 1,1157   4,484   399,3466   3465,3481

### X(3484) = INTERSECTION OF LINES X(3)X(2120) and X(4)X(54)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[4x2 - 5y2 - 5z2 + 2xy + 2xz + 10yz - (y + z)(3y + z)(y + 3z)/(2x + y + z)],
where x : y : z = 1/(b2 + c2 - a2) : 1/(c2 + a2 - b2) : 1/(a2 + b2 - c2)         (M. Iliev, 10/27/08)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3484) is the tangential of X(4) on the Neuberg cubic.

X(3484) lies on the Neuberg cubic and these lines: 3,2120   4,54   74,1157   484,3466   3479,3480

### X(3485) = INTERSECTION OF LINES X(1)X(4) AND X(7)X(21)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos A + cos B + cos C + cos B cos C
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3485) lies on these lines: 1,4   2,65   7,21   8,12   11,938   20,1836   29,1118   46,631   55,411   57,1125   78,2550   85,350   196,1940   221,940   238,1451   277,2140   329,958   354,1858   376,1770   443,997   459,1482   551,1420   608,2303   908,2551   945,1065   986,1393   1038,2263   1159,1656   1468,1935   1854,2883   2171,2345

X(3485) = X(968)-cross conjugate of X(966)

### X(3486) = INTERSECTION OF LINES X(1)X(4) AND X(8)X(21)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos A + cos B + cos C - cos B cos C
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3486) lies on these lines: 1,4   2,1837   3,1788   8,21   11,2476   20,65   25,1610   29,1857   46,376   78,2551   80,498   90,1000   144,145   281,284   452,960   519,1697   529,2098   1159,1657   1468,1936   1503,1854   2268,2345

X(3486) = reflection of X(i) in X(j) for these (i,j): (8,958), (388,1)
X(3486) = extangents-to-intangents similarity image of X(8)

### X(3487) = INTERSECTION OF LINES X(1)X(4) AND X(3)X(7)

Trilinears    1 + cos A + cos B + cos C + cos B cos C : :

Let A' be the midpoint of X(1) and the A-intouch point. Define B' and C' cyclically. The triangle A'B'C' is homothetic to the 2nd extouch triangle, and the center of homothety is X(3487). (Randy Hutson, September 14, 2016)

X(3487) lies on these lines: 1,4   2,72   3,7   5,938   8,442   9,1125   57,631   78,443   142,936   281,1148   329,405   386,2140   474,1260

### X(3488) = INTERSECTION OF LINES X(1)X(4) AND X(8)X(405)

Trilinears    1 + cos A + cos B + cos C - cos B cos C

X(3488) lies on these lines: 1,4   3,938   8,405   9,519   20,942   35,1788   57,376   387,1104   952,954

X(3488) = reflection of X(1056) in X(1)

### X(3489) = ISOGONAL CONJUGATE OF X(627)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/x(627)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3489) lies on the Napoleon cubic and these lines: 3,1337   5,302   51,61

X(3489) = isogonal conjugate of X(627)
X(3489) = X(54)-cross conjugate of X(3490)

### X(3490) = ISOGONAL CONJUGATE OF X(628)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/x(628)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3490) lies on the Napoleon cubic and these lines: 3,1338   5,303   51,62

X(3490) = isogonal conjugate of X(628)
X(3490) = X(54)-cross conjugate of X(3489)

### X(3491) = INTERSECTION OF LINES X(1)X(295) AND X(4)X(69)

Barycentrics   a2[a4b4 + a4c4 - a2(b6 + c6) + b2c2(b4 + c4)] : :      (E. Danneels)

X(3491) lies on the cubic Z(X(384)) and these lines: 1,295   4,69   32,1613   39,695   147,185   211,754   626,2387   3494,3497

X(3491) = X(384)-Ceva conjugate of X(39)

### X(3492) = INTERSECTION OF LINES X(4)X(83) AND X(32)X(695)

Barycentrics    a2(a8 - a4b2c2 - a2b6 - a2c6 - 2b4c4) : :      (E. Danneels)

X(3492) lies on the cubic Z(X(384)) and these lines: 1,3497   4,83   32,695   39,1915   184,194   3500,3502

X(3492) = X(384)-Ceva conjugate of X(32)

### X(3493) = INTERSECTION OF LINES X(4)X(194) AND X(32)X(694)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a6 - b6 - c6 + a2b2c2)/(a4 - b2c2)      (E. Danneels)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(3493) lies on the cubic Z(X(384)) and these lines: 4,147   32,694

### X(3494) = INTERSECTION OF LINES X(1)X(87) AND X(3)X(2053)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 + b2 + c2 - ab - ac + bc)/(ab + ac - bc)     (E. Danneels)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3494) lies on the cubic Z(X(384)) and these lines: 1,87   3,2053   32,2319   384,3502   3491,3501

X(3494) = isogonal conjugate of X(3502)

### X(3495) = INTERSECTION OF LINES X(32)X(983) AND X(39)X(256)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/[b2c2 + c2a2 + a2b2 - abc(b + c - a)]     (E. Danneels)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3495) lies on the cubic Z(X(384)) and these lines: 1,3499   32,983   39,256   384,3503   3498,3501

X(3495) = isogonal conjugate of X(3503)

### X(3496) = INTERSECTION OF LINES X(1)X(32) AND X(4)X(9)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b3 + c3 - a3 + abc      (E. Danneels)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3496) lies on the cubic Z(X(384)) and these lines: 1,32   4,9   6,986   39,893   57,348   63,194   191,2795   220,2943   257,384   517,2329   910,960   2170,2975   2288,2323

X(3496) = isogonal conjugate of X(3497)
X(3496) = crossdifference of every pair of points on the line X(1459)X(1491)

### X(3497) = ISOGONAL CONJUGATE OF X(3496)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1\(b3 + c3 - a3 + abc)     (E. Danneels)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3497) lies on the cubic Z(X(384)) and these lines: 1,3492   3,984   63,2896   257,384   607,2221   3491,3494

X(3497) = isogonal conjugate of X(3496)
X(3497) = X(172)-cross conjugate of X(1)

### X(3498) = INTERSECTION OF LINES X(3)X(83) AND X(194)X(263)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[2a6(b2 + c2) + a4(b4 + 3b2c2 + c4) + a2b2c2(b2 + c2) + b4c4]/(a4 - a2b2 - a2c2 - 2b2c2)     (E. Danneels)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3498) lies on the cubic Z(X(384)) and these lines: 3,83   3495,3501

### X(3499) = INTERSECTION OF LINES X(3)X(695) AND X(6)X(76)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[a4(b4 + b2c2 + c4) + a2b2c2(b2 + c2) - b4c4]     (E. Danneels)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3499) lies on the cubic Z(X(384)) and these lines: 3,695   6,76   2076,2916

X(3499) = X(i)-Ceva conjugate of X(j) for these (i,j): (384,3), (3051,6)
X(3499) = vertex conjugate of PU(141)

### X(3500) = INTERSECTION OF LINES X(1)X(295) AND X(3)X(238)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a2b + a2c - a(b2 + bc + c2) + bc( b + c)]     (E. Danneels)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3500) lies on the cubic Z(X(384)) and these lines: 1,295   3,238   63,194   222,1424   384,3501   3492,3502

X(3500) = isogonal conjugate of X(3501)
X(3500) = trilinear pole of line X(659)X(1459)
X(3500) = cevapoint of X(649) and X(3123)
X(3500) = X(2275)-cross conjugate of X(1)

### X(3501) = INTERSECTION OF LINES X(1)X(39) AND X(4)X(9)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2b - ab2 - abc - ac2 + a2c + bc2 + b2c
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3501) lies on the cubic Z(X(384)) and these lines: 1,39   2,1334   ,3,2329   4,9   7,979   8,672   32,3494   37,986   41,100   43,213   57,345   63,2896   76,1423   145,1475   220,1376   321,2198   346,1400   384,3500   404,644   484,1759   579,2321   1575,2176

X(3501) = isogonal conjugate of X(3500)
X(3501) = crosssum of X(649) and X(3123)
X(3501) = crossdifference of every pair of points on the line X(659)X(1459)

### X(3502) = ISOGONAL CONJUGATE OF X(3494)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (ab + ac - bc)/(a2 + b2 + c2 - ab - ac + bc)      (E. Danneels)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3502) lies on the cubic Z(X(384)) and these lines: 1,2896   39,3496   384,3494   3492,3500   3493,3503

X(3502) = isogonal conjugate of X(3494)

### X(3503) = ISOGONAL CONJUGATE OF X(3495)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [b2c2 + c2a2 + a2b2 - abc(b + c - a)]/(b + c - a)     (E. Danneels)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3503) lies on the cubic Z(X(384)) and these lines: 1,3   76,3501   83,3496   109,1923   384,3495   3493,3502

X(3503) = isogonal conjugate of X(3495)

### X(3504) = INTERSECTION OF LINES X(25)X(385) AND X(32)X(1613)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 - a2)/(b2c2 - c2a2 - a2b2)     (E. Danneels)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3504) lies on the cubic Z(X(385)) and these lines: 25,385   32,3229   1423,1716

X(3504) = isogonal cojugate of X(3186)
X(3504) = X(69)-Ceva conjugate of X(3)
X(3504) = X(2998)-cross conjugate of X(3224)

### X(3505) = INTERSECTION OF LINES X(32)X(695) AND X(98)X(783)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b6 + c6 - a6 - a2b2c2)/(a4 + b2c2)     (E. Danneels)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3505) lies on the cubic Z(X(385)) and these lines: 2,3186   98,783   3510,3512

### X(3506) = INTERSECTION OF LINES X(2)X(98) AND X(32)X(694)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a8 - a4b2c2 - a2b6 - a2c6 + 2b4c4)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3506) lies on the cubic Z(X(385)) and these lines: 1,3512   2,98   32,694   648,1974   1196,1915

X(3506) = X(385)-Ceva conjugate of X(32)

### X(3507) = INTERSECTION OF LINES X(1)X(2) AND X(32)X(2319)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(b + c) - (a2 + bc)(b2 + bc + c2) + a(b + c)(b2 + c2)      (E. Danneels)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3507) lies on the cubic Z(X(385)) and these lines: 1,2   32,2319   511,3512   694,3508   3225,3510

X(3507) = X(385)-Ceva conjugate of X(3508)

### X(3508) = INTERSECTION OF LINES X(1)X(6) AND X(76)X(1423)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 + bc)(b2 + bc + c2) - (b + c)(b2c2 + a2b2 + a2c2)     (E. Danneels)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3508) lies on the cubic Z(X(385)) and these lines: 1,6   32,983   76,1423   98,813   190,1966   256,1500   385,3512   694,3510   1018,1756   1334,1655

X(3508) = X(385)-Ceva conjugate of X(3507)

### X(3509) = INTERSECTION OF LINES X(1)X(32) AND X(2)X(7)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3 - b3 - c3 + abc
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3509) lies on the cubic Z(X(385)) and these lines: 1,32   2,7   6,982   19,2319   65,2329   98,813   101,758   171,846   199,228   291,1757   295,511   335,385   484,1018   522,649   529,1146   594,2160   666,2311   902,968   910,1282   1054,1575

X(3509) = isogonal cojugate of X(3512)
X(3509) = X(i)-Ceva conjugate of X(j) for these (i,j): (335,1), (385,3511)
X(3509) = crosssum of X(i) and X(j) for these (i,j): (659,2170), (2238,2292)
X(3509) = crossdifference of every pair of points on the line X(663)X(1193)

### X(3510) = INTERSECTION OF LINES X(1)X(76) AND X(6)X(43)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b3c3 - a3b3 - a3c3 + a2b2c2     (E. Danneels)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3510) lies on the cubic Z(X(385)) and these lines: 1,76   6,43   42,894   291,511   292,2238   1402,1423   1492,1933   3225,3507   3505,3512

X(3510) = X(i)-Ceva conjugate of X(j) for these (i,j): (385,3509), (1911,1)

### X(3511) = INTERSECTION OF LINES X(3)X(76) AND X(6)X(694)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a6b4 - a4b6 - a4c6 + a6c4 + b4c6 + b6c4 - a2b2c6 + a2b4c4 - a2b6c2 - a4b2c4 - a4b4c2 + a6b2c2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3511) lies on the cubic Z(X(385)) and these lines: 3,76   6,694   147,446   237,385

X(3511) = X(i)-Ceva conjugate of X(j) for these (i,j): (237,3), (385,6)
X(3511)= crosssum of X(523) and X(2679)
X(3511) = crossdifference of every pair of points on the line X(804)X(2023)

### X(3512) = ISOGONAL CONJUGATE OF X(3509)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/(a3 - b3 - c3 + abc)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3512) lies on the cubic Z(X(385)) and these lines: 1,3506   6,2114   19,1423   55,846   335,385   511,3507   1086,2160   1654,2893   3505,3510

X(3512) = isogonal cojugate of X(3509)
X(3512) = cevapoint of X(i) and X(j) for these (i,j): (659,2170), (2238,2292)
X(3512) = X(1914)-cross conjugate of X(1)
X(3512)= crosssum of X(846) and X(1282)

### X(3513) = 1st DILATION CENTER

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b + c)(a + b - c) + 4r(r2 + 4rR)1/2 (Peter Moses, 7/22/08)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(3513) = (r + 2R + sqrt(r2 + 4rR))*X(1) - 2r*X(3)

For any two points P, Q in the set {A, B, C} and arbitrary point X, let M(X) = (|PX| + |QX|)/|PQ|. The point X that minimizes M(X) is X(3513). See

David Eppstein, The dilation center of a triangle

for a discussion of X(3513) as a point of intersection of three similar ellipses, each having two of the points A, B, C as foci. (Contributed by David Eppstein, 7/10/08)

X(3513) is the point P such that the incenter of the circumcevian triangle of P is also the incenter of triangle ABC. (Randy Hutson, 9/23/2011)

X(3513) and X(3514) are the limiting points (point-circles) of the coaxal system that includes the circumcircle and incircle. Their midpoint, X(3660), is the radical trace of the incircle and circumcircle. (Peter Moses, November 15, 2011)

If you have The Geometer's Sketchpad, you can view X(3513).

X(3513) lies on this line: 1,3

X(3513) = reflection of X(3514) in X(3660)
X(3513) = inverse-in-circumcircle of X(3514)
X(3513) = inverse-in-incircle of X(3514)
X(3513) = one of two harmonic traces of the Soddy circles; the other is X(3514)
X(3513) = {X(i),X(j)}-harmonic conjugate of X(3514) for these (i,j): (1,57), (3,1617), (36,2078), (55,56), (65,354), (165,1420)

### X(3514) = INVERSE-IN-CIRCUMCIRCLE OF X(3513)

Trilinears    (a - b + c)(a + b - c) - 4r(r2 + 4rR)1/2 : : (Peter Moses, 7/24/08)
X(3514) = (r + 2R - sqrt(r2 + 4rR))*X(1) - 2r*X(3)

Like X(3513), the point X(3514) is the point of intersection of three similar ellipses, each having two of the points A, B, C as foci. (Contributed by Peter Moses, 7/24/08)

If you have The Geometer's Sketchpad, you can view X(3514).

X(3514) lies on this line: 1,3

X(3514) = reflection of X(3513) in X(3660)
X(3514) = inverse-in-circumcircle of X(3513)
X(3514) = inverse-in-incircle of X(3513)
X(3514) = one of two harmonic traces of the Soddy circles; the other is X(3513)
X(3514) = {X(i),X(j)}-harmonic conjugate of X(3513) for these (i,j): (1,57), (3,1617), (36,2078), (55,56), (65,354), (165,1420)

### X(3515) = EULER LINE INTERCEPT OF LINE X(154)X(185)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 3 cos A - sec A
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(3515) has Shinagawa coefficients (3F, -E - 3F).

X(3515) lies on these lines: 2,3   36,1398   64,1620   154,185   159,2929   165,1902   187,2207   232,3053   1033,1609   1204,1495   1350,1974   1452,2646

X(3515) = crosspoint of X(250) and X(1301)
X(3515) = homothetic center of Kosnita triangle and mid-triangle of orthic and circumorthic triangles

### X(3516) = EULER LINE INTERCEPT OF LINE X(64)X(184)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 3 cos A + sec A
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(3516) has Shinagawa coefficients (3F,E - 3F).

X(3516) lies on these lines: 2,3   6,1204   393174   51,1192   55,1398   64,184   165,1829   574,2207   1181,3357

X(3516) = homothetic center of Trinh triangle and mid-triangle of orthic and circumorthic triangles

### X(3517) = EULER LINE INTERCEPT OF LINE X(154)X(389)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 3 cos A - 2 sec A
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(3517) has Shinagawa coefficients (3F, -2E - 3F).

X(3517) lies on these lines: 2,3   52,3167   154,389   231,1609   1147,1351   1181,1495   1384,2207   3053,3199   3357,3426

X(3517) = homothetic center of tangential triangle and mid-triangle of orthic and circumorthic triangles

### X(3518) = EULER LINE INTERCEPT OF LINE X(51)X(54)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 4 cos A - 3 sec A
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(3518) has Shinagawa coefficients (4F, -3E - 4F).

X(3518) lies on these lines: 2,3   49,143   51,54   52,110   53,1601   93,324   98,3456   107,1141   112,3199   156,568   184,1199   389,1495   575,1843   576,1974   1112,3043   1147,3060   1304,3470   1866,1870

X(3518) = isogonal conjugate of X(3519)
X(3518) = X(1179)-Ceva conjugate of X(4)
X(3518) = X(2965)-cross conjugate of X(1994)
X(3518) = crosspoint of X(3442) and X(3443)
X(3518) = crosssum of X(633) and X(634)
X(3518) = X(3)-isoconjugate of X(2962)

### X(3519) = ISOGONAL CONJUGATE OF X(3518)

Trilinears    1/(4 cos A - 3 sec A) : :
Trilinears    csc A tan 3A + sec 3A : :
Barycentrics    1/(cot A - 3 tan A) : :
Barycentrics    1/((3 S^2-SA^2) SB SC) : :
Barycentrics    (a^2 - b^2 - c^2)*(a^4 - a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4) : :

Let A'B'C' be the reflection triangle. Let Oa be the circle centered at A' and passing through A, and define Ob and Oc cyclically. The radical center of Oa, Ob, Oc is X(3519). Let Na be the reflection of X(5) in the perpendicular bisector of BC, and define Nb and Nc cyclically. The lines ANa, BNb, CNc concur in X(3519). Let A''B''C'' be the Trinh triangle. Let A* be the orthopole of line B'C', and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(3519). (Randy Hutson, October 13, 2015)

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26850.

X(3519) lies on the the Jerabek hyperbola, the cubic K618, and these lines:
{2, 1493}, {3, 539}, {4, 93}, {5, 1173}, {6, 17}, {20, 13452}, {49, 343}, {54, 140}, {64, 1657}, {65, 2962}, {68, 12606}, {69, 12363}, {70, 14791}, {74, 550}, {185, 14861}, {265, 5562}, {290, 7768}, {340, 8795}, {394, 15317}, {399, 14862}, {511, 13433}, {524, 5576}, {542, 13564}, {567, 12899}, {895, 11585}, {1176, 3564}, {1503, 9935}, {1899, 11577}, {2889, 3448}, {2914, 14643}, {2918, 5898}, {2979, 12291}, {3426, 5073}, {3431, 3523}, {3521, 13754}, {3522, 11270}, {3527, 3574}, {3850, 14483}, {4857, 6286}, {5056, 13565}, {5059, 11738}, {5068, 14491}, {5449, 15002}, {5504, 12359}, {6101, 12226}, {6145, 7574}, {6515, 9827}, {7517, 15069}, {10018, 11597}, {10625, 14864}, {10628, 11744}, {11138, 11600}, {11139, 11601}, {11225, 15047}, {11412, 12280}, {12936, 15232}

X(3519) = midpoint of X(i) and X(j) for these {i,j}: {2888, 12325}, {11412, 12280}
X(3519) = reflection of X(i) in X(j) for these {i,j}: {195, 1209}, {6243, 6152}, {6288, 2888}, {11271, 1493}, {12226, 6101}, {12316, 3574}, {13431, 12242}, {13432, 13431}
X(3519) = complement of X(11271)
X(3519) = anticomplement of X(1493)
X(3519) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1487, 8}, {2962, 2889}
X(3519) = cevapoint of X(i) and X(j) for these (i,j): {633, 634}, {14813, 14814}
> X(3519) = X(11140)-Ceva conjugate of X(2963)
X(3519) = X(1216)-cross conjugate of X(3)
X(3519) = barycentric product X(i)X(j) for these {i,j}: {3, 11140}, {63, 2962}, {69, 2963}, {93, 394}, {252, 343}, {525, 930}
X(3519) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 1994}, {5, 14129}, {6, 3518}, {17, 472}, {18, 473}, {48, 2964}, {51, 14577}, {69, 7769}, {93, 2052}, {184, 2965}, {216, 143}, {252, 275}, {562, 14165}, {570, 6152}, {577, 49}, {647, 1510}, {930, 648}, {2962, 92}, {2963, 4}, {8603, 10633}, {8604, 10632}, {11140, 264}, {14111, 11547}
X(3519) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3518}, {4, 2964}, {19, 1994}, {49, 158}, {92, 2965}, {143, 2190}, {162, 1510}, {1973, 7769}, {2148, 14129}, {2167, 14577}, {2216, 6152}
X(3519) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11271, 1493), (17, 18, 2963), (93, 562, 14111), (195, 1656, 12242), (195, 13432, 13431), (1209, 12242, 1656), (1209, 13431, 12242), (1656, 13432, 195), (12242, 13431, 195)

### X(3520) = EULER LINE INTERCEPT OF LINE X(54)X(74)

Trilinears    4 cos A + sec A : :

As a point on the Euler line, X(3520) has Shinagawa coefficients (4F, E - 4F).

X(3520) lies on these lines: 2,3    35,1870   39,112   49,3043   54,74   64,3431   93,930   99,1235   184,3357   567,1986   574,1968   578,1199   1970,3269

X(3520) = reflection of X(4) in X(1594)
X(3520) = homothetic center of circumorthic triangle and Trinh triangle
X(3520) = X(35)-of-Trinh-triangle if ABC is acute
X(3520) = Trinh-isogonal conjugate of X(7691)
X(3520) = exsimilicenter of circumcircle and Trinh circle; the insimilicenter is X(2071))

### X(3521) = ISOGONAL CONJUGATE OF X(3520)

Trilinears    1/(4 cos A + sec A) : :

Let A'B'C' be the orthocentroidal triangle. Let A" be the isogonal conjugate, wrt the A-altimedial triangle, of A'. Define B" and C" cyclically. The lines AA", BB", CC" concur in X(3521). (Randy Hutson, November 2, 2017)

Let Pa be the isotomic conjugate of X(1105) with respect to the triangle AX(3)X(4), and define Pb and Pc cyclically. Then PaPbPc and ABC are perspective at X(3521). (Angel Montesdeoca, September 23, 2018)

X(3521) lies on these lines: 3,1568   5,74   6,382   20,3431   30,54   64,381   185,265   597,1177

X(3521) = X(1531)-cross conjugate of X(265)

### X(3522) = EULER LINE INTERCEPT OF LINE X(8)X(165)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 3 cos A - cos B cos C
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sec A - 4 sec B sec C
Trilinears        h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = 4 sec B sec C - 3 sin B sin C
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(3522) has Shinagawa coefficients (3, -4).

X(3522) is one of many points on the Euler line that have a certain convenient representation, received from Peter Moses, February 8, 2010. Specifically, the appearance of (t,k) in the following list means that

X(k) = f(A,B,C) : f(B,C,A): f(C,A,B),

where f(A,B,C) = t cos A + cos B cos C:

(-4,3528), (-3,3522), (-5/2, 548), (-2,376), (-3/2,550), (-5/4,3534), (-1,20), (-3/4,1657), (-2/3,3529), (-1/3,3146), (-1/4,382), (-1/5,3543), (0,4), (1/6,546), (1/4,381), (1/3,3091), (2/5, 3545), (4/9,3544), (1/2,5), (2/3,3090), (7/10,547), (3/4,1656), (1,2), (7/6,632), (6/5,3533), (5/4,3526), (4/3,3525), (3/2,140), (2,631), (5/2,549), (3,3523), (7/2,3530), (4,3525), (infinity,3)

Other recent results on representations of points on the Euler line are given in

Peter J. C. Moses and Clark Kimberling, "Perspective isoconjugate triangle pairs, Hofstadter pairs, and crosssums on the nine-point circle," Forum Geometricorum 11 (2011) 83-93. Click here to download a pdf.

X(3522) lies on these lines: 2,3   8,165   40,145   56,390   78,144   84,3219   97,3346   185,2979   193,1350   323,1181   489,1270   490,1271   590,1131   615,1132   1038,3100   1155,3486   1193,1742   1204,3098   2646,3474

X(3522) = midpoint of X(20) and X(3091)
X(3522) = reflection of X(i) in X(j) for these (i,j): (4,1656), (631,3), (3091,631)
X(3522) = anticomplement of X(3091)
X(3522) = Thomson-isogonal conjugate of X(5644)
X(3522) = homothetic center of 2nd Conway triangle and cross-triangle of excentral and 2nd circumperp triangles

### X(3523) = EULER LINE INTERCEPT OF LINE X(35)X(390)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 3 cos A + cos B cos C
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sec A + 3 sec B sec C
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(3523) has Shinagawa coefficients (3, -2).

X(3523) lies on these lines: 2,3   35,390   36,3085   40,3306   84,3305   95,253   145,1385   147,620   153,2551   165,962   182,193   389,2979   391,572   487,1270   488,1271   991,3216   1151,3069   1152,3068   1155,3485   1788,2646   1189,2888

X(3523) = midpoint of X(i) in X(j) for these (i,j): (3,3526), (3090,3528)
X(3523) = reflection of X(i) in X(j) for these (i,j): (3090,3526), (3528,3)
X(3523) = anticomplement of X(3090)
X(3523) = homothetic center of dual of orthic triangle and (mid-triangle of orthic and circumorthic triangles)

### X(3524) = EULER LINE INTERCEPT OF LINE X(35)X(1058)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 4 cos A + cos B cos C
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 3 cos A + sin B sin C
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(3524) has Shinagawa coefficients (4, -3).

X(3524) lies on these lines: 2,3   35,1058   36,1056   40,551   69,3431   98,2482   182,1992   187,1285   519,3158   553,3487   597,1350   943,1466   1000,1319   1385,3241   3058,3086   3070,3316   3071,3317

X(3524) = midpoint of X(376) in X(3545)
X(3524) = reflection of X(i) in X(j) for these (i,j): (4,3545), (3545,2)
X(3524) = isogonal conjugate of X(3531)

### X(3525) = EULER LINE INTERCEPT OF LINE X(69)X(575)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 4 cos A + 3 cos B cos C
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 3 sec A + 4 sec B sec C
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(3525) has Shinagawa coefficients (4, -1).

X(3525) lies on these lines: 2,3   69,575   394,1199   498,1056   499,1058   944,1698   1007,1078   1285,2548   1587,3316   1588,3317   3053,3055   3085,3304   3086,3303

X(3525) = crosspoint of X(3316) and X(3317)
X(3525) = crosssum of X(3311) and X(3312)

### X(3526) = EULER LINE INTERCEPT OF LINE X(49)X(182)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 5 cos A + 4 cos B cos C
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 4 sec A + 5 sec B sec C
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(3526) has Shinagawa coefficients (5, -1).

X(3526) lies on these lines: 2,3   6,3411   49,182   143,2979   195,394   302,628   303,627   498,999   499,3295   567,1092   568,1216   575,599   590,3312   615,3311   1125,1482   1159,1788   1384,2548   1385,1698   1506,3053

X(3526) = midpoint of X(3090) and X(3523)
X(3526) = reflection of X(3) in X(3523)
X(3526) = complement of X(3090)
X(3526) = {X(3411),X(3412)}-harmonic conjugate of X(6)
X(3526) = homothetic center of medial triangle and mid-triangle of Euler and anti-Euler triangles
X(3526) = homothetic center of X(5)-altimedial and X(2)-anti-altimedial triangles

### X(3527) = ISOGONAL CONJUGATE OF X(631)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(cos A + sin B sin C)
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 1/(sec A + 2 sec B sec C)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3527) lies on these lines: 3,51   5,69   6,1598   24,3431   25,54   64,389   68,381   72,1482   73,999   74,1112   185,3426   1092,3066

X(3527) = X(1181)-cross conjugate of X(3)
X(3527) = isogonal conjugate of X(631)

### X(3528) = EULER LINE INTERCEPT OF LINE X(35)X(1056)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 5 cos A - sin B sin C
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sec A - 4 sec B sec C
Trilinears        h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = 5 cos B cos C - 4 sin B sin C
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(3528) has Shinagawa coefficients (4, -5).

X(3528) lies on these lines: 2,3   35,1056   36,1058   39,1285   40,3244   165,944   519,3161

X(3528) = reflection of X(i) in X(j) for these (i,j): (4,3090), (3090,3523), (3523,3)
X(3528) = inverse-in-orthocentroidal circle of X(3544)

### X(3529) = EULER LINE INTERCEPT OF LINE X(516)X(944)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 5 cos A - 3 sin B sin C
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 5 cos B cos C - 2 sin B sin C
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(3529) has Shinagawa coefficients (2, -5).

X(3529) lies on these lines: 2,3   516,944   1056,3303   1058,3304   1249,3284   1770,3486

X(3529) = reflection of X(i) in X(j) for these (i,j): (4,20), (20,1657), (382,550), (3146,3), (3543,3534)
X(3529) = anticomplement of X(382)
X(3529) = {X(4),X(3528)}-harmonic conjugate of X(2)
X(3529) = {X(382),X(550)}-harmonic conjugate of X(2)
X(3529) = Ehrmann-mid-to-Johnson similarity image of X(20)
X(3529) = homothetic center of cevian triangle of X(3) and mid-triangle of medial and anticomplementary triangles

### X(3530) = EULER LINE INTERCEPT OF LINE X(15)X(3411)

Trilinears  5 cos A + 2 sin B sin C : :
Trilinears    6a4 + b4 + c4 - 2b2c2 - 7a2b2 - 7a2c2 : :
X(3530) = 3X(2) + 5X(3) = 21X(2) - 5X(4) = 7X(3) + X(4)

As a point on the Euler line, X(3530) has Shinagawa coefficients (7, -5).

Let P7' and U7' be the bicentric pair constructed as with PU(7), but with circles half the size. The midpoint of P7' and U7' is X(3530). (Randy Hutson, December 26, 2015)

X(3530) lies on these lines: 2,3   15,3411   16,3412   1385,3244

X(3530) = midpoint of X(i) in X(j) for these (i,j): (3,140), (5,548), (546,550)
X(3530) = complement of X(546)

### X(3531) = INTERSECTION OF LINES X(3,373) AND X(69,381)

Trilinears    1/(3 cos A + sin B sin C) : :
Barycentrics   a^2/[7a^4 - 8a^2(b^2 + c^2) + (b^2 - c^2)^2] : :

X(3531) lies on the Jerabek hyperbola and these lines: 3,373   25,3431   51,3426   54,1598   69,381   74,1597   399,895

X(3531) = isogonal conjugate of X(3524)
X(3531) = cevapoint of X(11485) and X(11486)
X(3531) = perspector of circle (X(3),2R)

### X(3532) = INTERSECTION OF LINES X(6,1204) AND X(64,1620)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(4 cos A - 3 sin B sin C)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3532) lies on the Jerabek hyperbola and these lines: 4,1192   6,1204   64,1620   68,550   71,3207   72,165   74,1498   265,1657   378,1173   1181,3431   3357,3426

X(3532) = isogonal conjugate of X(3146)
X(3532) = cevapoint of X(1151) and X(1152)
X(3532) = X(154)-cross conjugate of X(6)

### X(3533) = EULER LINE INTERCEPT OF LINE X(371)X(3317)

Trilinears    6 cos A + 5 cos B cos C : :
Trilinears    5 sec A + 6 sec B sec C : :
Barycentrics    cot B cot C - 6 : :

As a point on the Euler line, X(3533) has Shinagawa coefficients (6, -1).

X(3533) lies on these lines: 2,3   371,3317   372,3316

### X(3534) = EULER LINE INTERCEPT OF LINE X(154)X(2777)

Trilinears     4 sec A - 5 sec B sec C : :
Trilinears     5 cos A - 4 cos B cos C : :
X(3534) = 2*X(2) - 3*X(3) = 2*X(381) - X(382)

As a point on the Euler line, X(3534) has Shinagawa coefficients (5, -9).

Let A'B'C' be the medial triangle. Let A" be the pole of line B'C' wrt the A-power circle, and define B", C" cyclically. X(3534) = X(2)-of-A"B"C". (Randy Hutson, March 14, 2018)

X(3534) lies on these lines: 2,3   154,2777   394,399   542,1350   590,1327   599,3098   615,1328   999,3058   1159,3474   1384,2549

X(3534) = midpoint of X(i) and X(j) for these (i,j): (20,376), (381,1657), (3529, 3543)
X(3534) = reflection of X(i) in X(j) for these (i,j): (3,376), (4,549), (376,550), (381,3), (382, 381), (549,548), (599,3098), (3543,5)
X(3534) = Stammler isogonal conjugate of X(6)
X(3534) = {X(2),X(3)}-harmonic conjugate of X(15693)
X(3534) = trisector nearest X(20) of segment X(3)X(20)

### X(3535) = EULER LINE INTERCEPT OF LINE X(275)X(3317)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec A + 2 csc A
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(3535) has Shinagawa coefficients (2F, S).

X(3535) lies on these lines: 2,3   275,3317   281,1659   393,590   587,1826   615,3087   1249,3068   1870,3083   2052,3316

X(3535) = inverse-in-orthocentroidal-circle of X(3536)
X(3535) = X(1151)-cross conjugate of X(1270)

### X(3536) = EULER LINE INTERCEPT OF LINE X(19)X(587)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec A - 2 csc A
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(3536) has Shinagawa coefficients (2F, -S).

X(3536) lies on these lines: 2,3   19,587   275,3316   393,615   590,3087   1249,3069   1870,3084   2052,3317

X(3536) = inverse-in-orthocentroidal-circle of X(3535)
X(3536) = X(1152)-cross conjugate of X(1271)

### X(3537) = EULER LINE INTERCEPT OF LINE X(1038)X(1058)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos B cos C + 2 sec B sec C
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(3537) has Shinagawa coefficients (2E, -2E + F).

X(3537) lies on these lines: 2,3   577,1285   1038,1058   1040,1056

### X(3538) = EULER LINE INTERCEPT OF LINE X(1038)X(1056)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos B cos C - 2 sec B sec C
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(3538) has Shinagawa coefficients (2E, -2E - F).

X(3538) lies on these lines: 2,3   1038,1056   1040,1058

### X(3539) = EULER LINE INTERCEPT OF LINE X(1056)X(3084)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos B cos C + 2 csc A
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(3539) has Shinagawa coefficients (2E, S).

X(3539) lies on these lines: 2,3   1056,3084   1058,3083

X(3539) = inverse-in-orthocentroidal-circle of X(3540)

### X(3540) = EULER LINE INTERCEPT OF LINE X(1056)X(3083)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos B cos C - 2 csc A
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(3540) has Shinagawa coefficients (2E, -S).

X(3540) lies on these lines: 2,3   1056,3083   1058,3084

X(3540) = inverse-in-orthocentroidal-circle of X(3539)

### X(3541) = EULER LINE INTERCEPT OF LINE X(33)X(499)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec A + 2 sin B sin C
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(3541) has Shinagawa coefficients (2F, E).

X(3541) lies on these lines: 2,3   33,499   34,498   54,66   254,264   317,1078   393,570   495,1398   571,3087   578,1899   590,3093   615,3092   847,1217   1092,1352   1199,2904   1870,3085   1968,2548

X(3541) = inverse-in-orthocentroidal-circle of X(3542)
X(3541) = anticomplement of X(3549)

### X(3542) = EULER LINE INTERCEPT OF LINE X(33)X(498)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec A - 2 sin B sin C
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(3542) has Shinagawa coefficients (2F, -E).

X(3542) lies on these lines: 2,3   33,498   34,499   70,1177   107,1299   158,1068   193,2904   206,1614   230,2207   254,1093   393,847   590,3092   615,3093   1300,1301   1352,1974   1870,3086   3186,3462

X(3542) = inverse-in-orthocentroidal-circle of X(3541)
X(3542) = anticomplement of X(3548)
X(3542) = X(1093)-Ceva conjugate of X(4)
X(3542) = cevapoint of X(155) and X(454)
X(3542) = X(155)-cross conjugate of X(4)

### X(3543) = EULER LINE INTERCEPT OF LINE X(146)X(148)

Trilinears    6 cos B cos C - sin B sin C : :
X(3543) = 5 X(2) - 4 X(3) = X(381) + 3*X(382)

As a point on the Euler line, X(3543) has Shinagawa coefficients (1, -6).

Let Pa be the parabola with focus A and directrix BC. Let La be the polar of X(3) wrt Pa. Define Lb and Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. Then X(3543) = X(193)-of-A'B'C'. (Randy Hutson, September 5, 2015)

X(3543) lies on these lines: 2,3   144,3419   146,148   147,543   152,544   153,528   253,317   371,1131   372,1132   388,3058   390,1478   393,3163   515,3241   519,962   541,3448   551,1699   553,938   671,2794   1503,1992

X(3543) = midpoint of X(2) and X(3146)
X(3543) = reflection of X(i) in X(j) for these (i,j): (2,4), (20,2), (376,381), (3529,3534), (3534,5)
X(3543) = anticomplement of X(376)
X(3543) = harmonic center of polar and de Longchamps circles
X(3543) = intersection of tangents to conic {{X(4),X(13),X(14),X(15),X(16)}} at X(13) and X(14)
X(3543) = {X(2),X(3)}-harmonic conjugate of X(15708)
X(3543) = {X(381),X(382)}-harmonic conjugate of X(15684)

### X(3544) = EULER LINE INTERCEPT OF LINE X(3070)X(3317)

Trilinears    5 cos B cos C + 4 sin B sin C : :
X(3544) = 8 X(3) + 9 X(4)

As a point on the Euler line, X(3544) has Shinagawa coefficients (4, 5).

X(3544) lies on these lines: 2,3   121,519   3070,3317   3071,3316

X(3544) = inverse-in-orthocentroidal-circle of X(3528)
X(3544) = crossdifference of every pair of points on the line X(647)X(2441)
X(3544) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,11737,3545), (3,4,11541), (5,5068,3545), (3090,3091,4)

### X(3545) = EULER LINE INTERCEPT OF LINE X(11)X(1056)

Trilinears    3 cos B cos C + 2 sin B sin C : :
Barycentrics    a^4 + 4a^2(b^2 + c^2) - 5(b^2 - c^2)^2 : :
X(3545) = 5 X(2) - 2 X(3) = 4 X(3) + 5 X(4)

As a point on the Euler line, X(3545) has Shinagawa coefficients (2, 3).

X(3545) lies on these lines: 2,3   11,1056   12,1058   69,1568   114,671   230,1285   262,538   355,3241   371,1328   372,1327   551,944   1131,3312   1132,3311   1352,1992   3058,3085

X(3545) = midpoint of X(4) and X(3524)
X(3545) = reflection of X(i) in X(j) for these (i,j): (376,3524), (3524,2)
X(3545) = inverse-in-orthocentroidal-circle of X(376)
X(3545) = inverse-in-circle-O(PU(5)) of X(382)
X(3545) = trisector nearest X(2) of segment X(2)X(4)
X(3545) = trisector nearest X(381) of segment X(2)X(381)
X(3545) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3,15702), (2,11737,3544), (5,5068,3544)

### X(3546) = EULER LINE INTERCEPT OF LINE X(498)X(1038)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec B sec C - 2 sin B sin C
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(3546) has Shinagawa coefficients (E - 2F, -E).

X(3546) lies on these lines: 2,3   498,1038   499,1040   590,1578   615,1579   1060,3085   1062,3086   1092,1899

X(3546) = complement of X(3089)

### X(3547) = EULER LINE INTERCEPT OF LINE X(69)X(155)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec B sec C + 2 sin B sin C
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(3547) has Shinagawa coefficients (E + 2F, -E).

X(3547) lies on these lines: 2,3   68,1176   69,155   343,1181   498,1040   499,1038   577,2548   590,1579   615,1578   1060,3086   1062,3085

### X(3548) = EULER LINE INTERCEPT OF LINE X(68)X(125)

Trilinears    sec B sec C - 4 sin B sin C : :

As a point on the Euler line, X(3548) has Shinagawa coefficients (E - 4F, -E).

Let La be the polar of X(4) wrt the circle centered at A and passing through X(3), and define Lb and Lc cyclically. (Note that X(4) is the perspector of any circle centered at a vertex of ABC.) Let A" = Lb∩Lc, and define B" and C" cyclically. Triangle A"B"C" is homothetic to the medial triangle, and the center of homothety is X(3548). (Randy Hutson, December 11, 2015)

X(3548) lies on these lines: 2,3   68,125   254,2970   498,1060   499,1062   1147,1899   1204,1568

X(3548) = complement of X(3542)

### X(3549) = EULER LINE INTERCEPT OF LINE X(68)X(184)

Trilinears    sec B sec C + 4 sin B sin C : :

As a point on the Euler line, X(3549) has Shinagawa coefficients (E + 4F, -E).

X(3549) lies on these lines: 2,3   68,184   155,343   193,195   206,1209   216,2165   498,1062   499,1060   577,1506

X(3549) = complement of X(3541)
X(3549) = homothetic center of orthocevian triangle of X(2) and Johnson triangle

### X(3550) = INTERSECTION OF LINES X(1)X(3) AND X(31)X(43)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2a2 + bc - ca - ab
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3550) lies on these lines: 1,3   2,902   31,34   32,2319   42,3097   63,3099   81,2177   87,2209   172,3208   200,1707   238,1376   595,978   609,1018   612,846   614,1054   750,1621   968,1961   983,1423   1120,2163   1740,1918   1955,2947   2308,3240   2329,3053

X(3550) = isogonal conjugate of X(3551)
X(3550) = X(983)-Ceva conjugate of X(1)
X(3550) = crosssum of X(513) and X(3123)

### X(3551) = ISOGONAL CONJUGATE OF X(3550)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/(2a2 + bc - ca - ab)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3551) lies on the Feuerbach hyperbola and these lines: 1,1463   8,726   9,1575   87,3123   983,1423   1149,2320

X(3551) = isogonal conjugate of X(3550)
X(3551) = X(982)-cross conjugate of X(1)

### X(3552) = EULER LINE INTERCEPT OF LINE X(32)X(99)

Trilinears    bc(2a4 + b2c2 - c2a2 - a2b2) : :

As a point on the Euler line, X(3552) has Shinagawa coefficients ((E + F)2 - 3S2, 4S2).

X(3552) lies on these lines: 2,3   32,99   69,2076   76,187   83,574   172,192   287,1204   330,1914   385,1975   1613,3360

X(3552) = cevapoint of X(3) and X(3360)
X(3552) = anticomplement of X(5025)
X(3552) = isogonal conjugate of perspector of 2nd Brocard circle

### X(3553) = INTERSECTION OF LINES X(1)X(6) AND X(19)X(41)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + sin A sin B sin C
Trilinears        aR + rs : bR + rs : cR + rs
Trilinears        a2bc + S2 : b2ca + S2 : c2ab + S2
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3553) lies on these lines: 1,6   19,41   33,42   34,3087   35,1609   48,2285   57,2178   65,198   71,1182   78,2345   165,1030   172,577   200,594   205,1474   216,1040   284,1766   326,894   380,584   571,609   610,2174   612,2318   800,1500   1490,1901   1834,2910   1953,2082   2099,2262   2266,2294   2270,3340   2303,2327   3068,3084   3069,3083

### X(3554) = INTERSECTION OF LINES X(1)X(6) AND X(19)X(604)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A - sin A sin B sin C
Trilinears        aR - rs : bR - rs : cR - rs
Trilinears        a2bc - S2 : b2ca - S2 : c2ab - S2
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3554) lies on these lines: 1,6   19,604   33,3087   34,393   36,1609   48,2082   56,2262   57,2164   198,319   216,1038   239,326   269,1086   282,1146   374,1696   380,2278   577,1040   800,1015   836,3086   1033,1398   1182,2260   1249,1870   1404,2261   1407,1422   1420,2178   1953,2285   3068,3083   3069,3084

X(3554) = crosspoint of X(1) and X(1422)
X(3554) = crosssum of X(i) and X(j) for these (i,j): (1,2324), (3083, 3084)

### X(3555) = DOSA POINT

Trilinears    bc(2a - b cos A - c cos A) : :

See Tibor Dosa, "Some triangle centers associated with excircles," Forum Geometricorum 7 151-158.

X(3555) lies on these lines: 1,6   8,443   10,354   20,145   28,1280   63,3295   65,519   78,999   200,474   210,1125   244,3214   319,2891   329,1058   379,3187   388,3419   496,908   528,1770   758,3057   912,1482   938,3421   962,971   997,3304   1009,2350   1320,2771   1376,3338   1858,2098   1872,1897   1998,3149   2093,2136

X(3555) = reflection of X(i) in X(j) for these (i,j): (8,942), (72,1), (3057,3244)
X(3555) = orthologic center of these triangles: Hutson-extouch to Hutson-intouch; the reciprocal orthologic center is X(5920)

### X(3556) = INTERSECTION OF LINES X(25)X(65) AND X(31)X(56)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(-aSBSC + bSCSA + cSASB)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3556) = X(188)-of-the-tangential-triangle (Peter Moses, 2/12/09)
X(3556) = X(65)-of-the-excentral-triangle-of-the-tangential-triangle (Peter Moses, 2/12/09)

X(3556) lies on these lines: 1,159   3,960   6,1245   20,1610   25,65   31,56   40,197   55,976   63,1619   64,71   73,2187   100,1265   161,2099   222,1660   859,1780   958,1503   1012,2217   1398,1456   1498,3428   1593,2182   2176,2178

X(3556) = X(63)-Ceva conjugate of X(6)
X(3556) = crosssum of X(i) and X(j) for these (i,j): (123,522), (513,2968)
X(3556) = pole wrt circumcircle of line X(521)X(656)

### X(3557) = 1st PAPPUS POINT

Trilinears    e sin A + sin(A - ω) : : , where e is defined at X(1340)
Trilinears        g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a(a2 + Z), where Z2 = a4 + b4 + c4 - b2c2 - c2a2 - a2b2
Barycentrics   ag(a,b,c) : bg(b,c,a) : cg(c,a,b)
X(3557) = 2X(3) - 3X(1340)
X(3557) = 4X(39) - 3X(1341)

X(3557) was submitted and called the Pappus point by Roland Bacher (March 11, 2009); coordinates and the related point X(3558) were found by Peter Moses (March 12, 2009).

X(3557) is the crosspoint (and crosssum) of the real foci of the Steiner inellipse. (Bernard Gibert, January 4, 2015)

X(3557) is the Brocard axis intercept, other than X(1340), of the circle {{X(1340),PU(1)}}. Also, X(3557) is the insimilicenter of the 2nd Lemoine circle and the circle {{X(371),X(372),PU(1),PU(39)}}. (Randy Hutson, January 5, 2015)

X(3557) lies on the cubics K028, K289, K704 and these lines: 3,6   4,3414   194,3413   262,6178

X(3557) = reflection of X(i) in X(j) for these (i,j): (1379,2029), (3558,3095)
X(3557) = isogonal conjugate of X(6177)
X(3557) = inverse-in-2nd-Brocard-circle of X(1341)
X(3557) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,576,3558), (6,32,3558), (6,1379,1341), (6,2029,1340), (61,62,3558), (371,372,1380), (1670,1671,1341), (1687,1688,2558), (1689,1690,3558), (3104,3105,3558)
X(3557) = crosssum of PU(118)
X(3557) = crosspoint of PU(118)

### X(3558) = 2nd PAPPUS POINT

Trilinears    e sin A - sin(A - ω) : :, where e is defined at X(1340)
Trilinears    a(a2 - Z), where Z2 = a4 + b4 + c4 - b2c2 - c2a2 - a2b2
X(3558) = 2X(3) - 3X(1341)
X(3558) = 4X(39) - 3X(1340)

X(3558) is the crosspoint (and crosssum) of the imaginary foci of the Steiner inellipse. (Bernard Gibert, January 4, 2015)

X(3558) is the Brocard axis intercept, other than X(1341), of the circle {{X(1341),PU(1)}}. Also, X(3558) is the exsimilicenter of the 2nd Lemoine circle and the circle {{X(371),X(372),PU(1),PU(39)}}. (Randy Hutson, January 5, 2015)

X(3558) lies on the cubics K028, K289, K704 and these lines: 3,6   4,3413   194,3414   262,6177

X(3558) = reflection of X(i) in X(j) for these (i,j): (1380,2028), (3557,3095)
X(3558) = isogonal conjugate of X(6178)
X(3558) = inverse-in-2nd-Brocard-circle of X(1340)
X(3558) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,576,3557), (6,32,3557), (6,1380,1340), (6,2028,1341), (61,62,3557), (371,372,1379), (1670,1671,1340), (1687,1688,2559), (1689,1690,3557), (3104,3105,3557)

### X(3559) = SS(sin A → cos A) OF X(333)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos B cos C)(cos B + cos C - cos A)/(cos B + cos C)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(3559) has Shinagawa coefficients (\$aSA\$F, \$aSBSC\$ + (\$aSA\$ - abc)F - \$a\$S2).

X(3559) lies on these lines: 2,3   58,1785   90,1896   158,920   162,1780   225,283   243,1858   318,333   393,1778   908,1819   1068,3157   1838,2328

X(3559) = X(1896)-Ceva conjugate of X(29)
X(3559) = cevapoint of X(i) and X(j) for these (i,j): (4,920), (46,1068), (453,3193)
X(3559) = X(i)-cross conjugate of X(j) for these (i,j): (46,3193), (3193,29)

### X(3560) = SS(sin A → cos A) OF X(940)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos2A + 2 cos B cos C + cos C cos A + cos A cos B
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(3560) has Shinagawa coefficients (\$aSA\$, 2abc - \$aSA\$), and also Shinagawa coefficients (R + r, R - r).

X(3560) lies on these lines: {1, 90}, {2, 3}, {35, 5587}, {40, 5251}, {55, 355}, {56, 5886}, {58, 5707}, {65, 920}, {84, 5436}, {100, 5818}, {104, 3616}, {119, 498}, {255, 2654}, {284, 5778}, {329, 5761}, {499, 1470}, {515, 5248}, {517, 958}, {573, 4877}, {581, 4653}, {938, 5770}, {943, 4313}, {944, 1621}, {946, 993}, {952, 3295}, {954, 5779}, {956, 1482}, {971, 1001}, {999, 3485}, {1125, 5450}, {1259, 3419}, {1698, 2077}, {1780, 5398}, {1898, 2646}, {2975, 5603}, {3576, 5259}, {3601, 5720}, {3746, 5881}, {3817, 5267}, {3929, 5258}, {4267, 5788}, {4654, 5563}, {5086, 5687}, {5260, 5657}, {5446, 5752}, {5698, 5762}, {5703, 5811}, {5744, 5804}, {5841, 6585}

X(3560) = complement of X(6850)

### X(3561) = SS(sin A → cos A) OF X(595)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)(cos2A - cos B cos C + cos C cos A + cos A cos B)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3561) lies on these lines: 1,21   3,1425   20,109   225,412   377,1771   1040,3215   1092,1813   1259,1331

X(3561) = crosssum of X(2310) and X(2501)

### X(3562) = SS(sin A → cos A) OF X(1621)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos2A - cos B cos C - cos C cos A - cos A cos B
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3562) lies on these lines: 1,21   4,651   6,938   8,394   20,222   40,77   65,775   73,411   100,1771   110,270   145,280   221,962   404,3075   416,820   950,2003   1066,3072   1071,3100   1406,3474   1895,3194   1935,2654

X(3562) = crosssum of X(647) and X(2310)

### X(3563) = 1st MOSES CIRCUMCIRCLE POINT

Trilinears    (sec A)/(b^2 cos^2 C + c^2 cos^2 B - bc cos A) : :
Trilinears    a/[(b2 + c2 - a2)(2a4 + b4 + c4 - 2b2c2 - c2 a2 - a2b2)] : :

The antipodal pair X(3563) to X(3565) were described by Peter Moses, Dec. 13, 2004.

Let A' be the reflection in line BC of the A-vertex of the antipedal triangle of X(6), and define B' and C' cyclically. Let OA be the circumcenter of AB'C', and define OB and OC cyclically. Let U be the circumcenter of A'BC, and define V and W cyclically. Let O' be the circumcenter of OAOBOC. Let O'' be the circumcenter of UVW. Then X(3563) = Λ(O',O''). Also, O'O''∩(infinity line) = X(3564). (Randy Hutson, June 19, 2015)

X(3563) lies on the circumcircle and these lines: 2,136   3,2971   4,99   24,112   25,110   100,1824   101,2333   107,459   186,691   232,1692   378,1296   403,935   427,930   468,476   934,1426

X(3563) = reflection of X(3565) in X(3)
X(3563) = isogonal conjugate of X(3564)
X(3563) = cevapoint of X(25) and X(232)
X(3563) = Λ(X(5), X(6))
X(3563) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {A,B,C,X(2),X(24)}
X(3563) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {A,B,C,X(4),X(25)}
X(3563) = inverse-in-polar-circle of X(114)
X(3563) = inverse-in-{circumcircle, nine-point circle}-inverter of X(136)
X(3563) = trilinear pole of line X(6)X(924)
X(3563) = Ψ(X(6), X(924))
X(3563) = Cundy-Parry Phi transform of X(14248)
X(3563) = Cundy-Parry Psi transform of X(6337)
X(3563) = perspector, wrt dual of orthic triangle, of polar circle

### X(3564) = 1st MOSES INFINITY POINT

Trilinears   (cos A)(b^2 cos^2 C + c^2 cos^2 B - bc cos A) : :
Trilinears   bc[(b2 + c2 - a2)(2a4 + b4 + c4 - 2b2c2 - c2a2 - a2b2)] : :

X(3564) and X(3566) are on the line at infinity. They may be regarded as directions in the plane of ABC, and as such are perpendicular. Continuing from X(3563), the infinite point of the line O'O'' is X(3564). (Randy Hutson, June 19, 2015)

X(3564) lies on these (parallel) lines: 2,3167   3,69   4,193   5,6   26,159   30,511   52,1843   98,325   110,468   114,230   115,1570   125,3292   140,141   147,385   156,206   184,343   265,895   287,441   323,858   381,1992   383,3181   394,1368   427,1993   428,3060   429,3193   495,611   496,613   546,576   547,597   548,3098   549,599   550,1350   1080,3180   1483,3242   1625,2211   1994,3410   2023,2025   2930,2931

X(3564) = isogonal conjugate of X(3563)
X(3564) = crossdifference of every pair of points on line X(6)X(924)
X(3564) = Cundy-Parry Phi transform of X(6337)
X(3564) = Cundy-Parry Psi transform of X(12248)

### X(3565) = 2nd MOSES CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[(3a2 - b2 - c2)(b2 - c2)]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let La be the reflection of the line X(5)X(6) in line BC, and define Lb and Lc cyclically. Let A' = Lc∩Lc, B' = Lc∩La, C' = La∩Lb. The lines AA', BB', CC' concur in X(3565). (Randy Hutson, January 29, 2018)

X(3565) lies on the circumcircle and these lines: 2,2374   3,2971   20,98   22,111   376,1300   378,1299   842,2071   858,2770   1370,2373 &nbnbsp; 2456,2698

X(3565) = reflection of X(3563) in X(3)
X(3565) = isogonal conjugate of X(3566)
X(3565) = crosssum of X(512) and X(2519)
X(3565) = X(2489)-cross conjugate of X(2)
X(3565) = anticomplement of X(5139)
X(3565) = trilinear pole of line X(6)X(1196)
X(3565) = Ψ(X(6), X(1196))
X(3565) = Λ(trilinear polar of X(459))

### X(3566) = 2nd MOSES INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(3a2 - b2 - c2)(b2 - c2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3566) lies on these (parallel) lines: 30,511   64,879   669,3265   1640,1853   2088,3143   2489,2506   2491,2524   2450,3569

X(3566) = isogonal conjugate of X(3565)
X(3566) = crossdifference of every pair of points on line X(6)X(1196)

### X(3567) = DON QUIXOTE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[a6(b2 + c2) - a4(3b4 + b2c2 + 3c4) + 3a2(b2 + c2)(b2 - c2)2 - (b4 -b2c2 + c4)(b2 - c2)2]

Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2[2S4 + SBSC(2S2 - SBSC - (1/2)(2 + b2 + c2)SASBSC

This point is the first of three Spanish Points developed by Antreas P. Hatzipolakis and Javier Garcia Capitan in 2009; see

The other two Spanish Points are the Sancho Panza Point X(3060) and the Miguel de Cervantes Point X(143).

X(3567) lies on these lines: 2,52   3,143   4,51   5,568   6,24   25,1614   74,1112   140,2979   155,1995   184,1199   186,578   511,631   567,1658   576,1092   1147,1994   1154,1656

X(3567) = crosssum of X(3) and X(1656)
X(3567) = homothetic center of orthocentroidal triangle and X(4)-adjunct anti-altimedial triangle
X(3567) = homothetic center of X(5)-altimedial and X(2)-adjunct anti-altimedial triangles
X(3567) = insimilicenter of the circumcircle and the nine-point circle of the orthic triangle. (The exsimilicenter is X(3060).)

### X(3568) = BELTRAMI-EULER POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[a12 - 2(b2 + c2)a10 + (b4 + 4b2c2 + c4)a8 - 2(b6 + c6)a6
+ (3b8 - 3b6c2 + b4c4 - 3b2c6 + 3c8)a4
- (b10 - b6c4 - b4c6 + c10)a2 + b4c4(b2 - c2)2]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

As a point on the Euler line, X(3568) has Shinagawa coefficients ((E + F)3F - (E + F)(E + 10F)S2 + 5S4, (E + F)4 - 2(E + F)(2E - 7F)S2 - 3S4).

Let P(2) and U(2) be the 1st and 2nd Beltrami points (as indexed at Bicentric Pairs, accessible using the Tables button at the top of ETC), and let P(40) and U(40) be the isogonal conjugates of P(2) and U(2), respectively. Then X(3568) is the point of intersection of the lines P(2)P(40) and U(2)U(40). The name "Beltrami-Euler Point" signifies the fact that X(3568) lies on the Euler line. See also X(2966) and X(3569). Contributed by Chris van Tienhoven, June 7, 2009.

X(3568) lies on this line: 2,3

### X(3569) = BELTRAMI-PARRY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 - c2)(b4 + c4 - a2b2 - a2c2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let P(2) and U(2) be the 1st and 2nd Beltrami points (as indexed at Bicentric Pairs, accessible using the Tables button at the top of ETC), and let P(40) and U(40) be the isogonal conjugates of P(2) and U(2), respectively. Then X(3569) is the point of intersection of the lines P(2)U(2) and P(40)U(40). The name "Beltrami-Parry Point" signifies the fact that X(3569) lies on the line X(74)X(111); where X(111) is the Parry point.and X(74) is the isogonal conjugate of the Euler infinity point. See also X(3568). Contributed by Chris van Tienhoven, June 7, 2009..

X(3569) lies on these lines: 6,526   74,111   110,112   113,1560   115,125   187,237   248,2422   297,525   520,2451   684,2491   688,2514   694,804   879,1987   924,2485   1499,1513   1510,3050   1769,2294   2450,3566

X(3569) = isogonal conjugate of X(2966)
X(3569) = reflection of X(i) in X(j) for these (i,j): (6,2492), (1640,1637), (2451,2489), (3049,2485), (3288,647)
X(3569) = X(i)-Ceva conjugate of X(j) for these (i,j): (297,868), (648,2967), (694,3124), (878,2451), (1987,3269), (2421,511), (2715,6)
X(3569) = crosspoint of X(i) and X(j) for these (i,j): (6,2515), (98,648), (110,2987)
X(3569) = crosssum of X(i) and X(j) for these (i,j): (2,2799), (98,2395), (112,2409), (230,523), (248,879), (441,525), (511,647)
X(3569) = crossdifference of every pair of points on the line X(2)X(98)
X(3569) = inverse-in-Parry-circle of X(647)
X(3569) = inverse-in-Parry-isodynamic-circle of X(1495); see X(2)
X(3569) = X(850)-of-1st-Brocard-triangle
X(3569) = X(6)-of-2nd-Parry-triangle
X(3569) = X(6)-of-3rd-Parry-triangle
X(3569) = intersection of trilinear polar of X(6) and polar wrt polar circle of X(6)
X(3569) = centroid of (degenerate) cross-triangle of 2nd and 3rd Parry triangles
X(3569) = center of the (degenerate) perspeconic of the 2nd and 3rd Parry triangles

### X(3570) = INTERSECTION OF LINES P(6)U(8) AND P(8)U(6)

Trilinears        bc(a2 - bc)/(b - c) : ca(b2 - ca)/(c - a) : ab(c2 - ab)/(a - b)
Barycentrics   (a2 - bc)/(b - c) : (b2 - ca)/(c - a) : (c2 - ab)/(a - b)

The points given by trilinears P(6) = b : c : a and U(6) = c : a : b are indexed at Bicentric Pairs (accessible using the Tables button at the top of ETC), and likewise for their isogonal conjugates P(8) = 1/b : 1/c : 1/a and U(8) = 1/c : 1/a : 1/b. The bicentric pair of lines P(6)U(8) and P(8)U(6) concur in X(3570). Contributed by Peter Moses, July 7, 2009..

If you have The Geometer's Sketchpad, you can view X(3570).

X(3570) lies on these lines: 2,6   100,190   101,668   106,3227   662,799   666,1026   1016,1023   1914,3253

X(3570) = isogonal conjugate of X(3572)
X(3570) = isotomic conjugate of X(4444)
X(3570) = X(666)-Ceva conjugate of X(190)
X(3570) = cevapoint of X(659) and X(2238)
X(3570) = crossdifference of every pair of points on the line X(512)X(1015)
X(3570) = trilinear pole of line X(238)X(239)

### X(3571) = INTERSECTION OF LINES P(6)P(8) AND U(6)U(8)

Trilinears    a4bc + a2(b4 - b3c - 2b2c2 - bc3 + c4) + b3c3 : :
Trilinears    (a + b)(c - a)(b2 + ac)(c2 - ab) - (a - b)(c + a)(b2 - ac)(c2 + ab) : :

The points given by trilinears P(6) = b : c : a and U(6) = c : a : b are indexed at Bicentric Pairs (accessible using the Tables button at the top of ETC), and likewise for their isogonal conjugates P(8) = 1/b : 1/c : 1/a and U(8) = 1/c : 1/a : 1/b. The bicentric pair of lines P(6)P(8) and U(6)U(8) concur in X(3571). Contributed by Peter Moses, July 7, 2009.

Continuing, the line P(6)P(8) is the tangent to the 1st bicentric of the Kiepert hyperbola at P(8), and line U(6)U(8) is the tangent to the 2nd bicentric of the Kiepert hyperbola at U(8). (Randy Hutson, March 25, 2016)

X(3571) lies on these lines: 1,512   9,43   1621,1964

X(3571) = crossdifference of PU(90)
X(3571) = crossdifference of every pair of points on line X(2238)X(4367)

### X(3572) = INTERSECTION OF LINES P(6)U(6) AND P(8)U(8)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b - c)/(a2 - bc)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

The points given by trilinears P(6) = b : c : a and U(6) = c : a : b are indexed at Bicentric Pairs (accessible using the Tables button at the top of ETC), and likewise for their isogonal conjugates P(8) = 1/b : 1/c : 1/a and U(8) = 1/c : 1/a : 1/b. The pair of central lines P(6)U(6) and P(8)U(8) concur in X(3572). Contributed by Peter Moses, July 7, 2009.

X(3572) lies on these lines: 2,661   6,798   37,513   42,649   111,741   291,1635   335,812   660,1026   813,901   1019,2084

X(3572) = isogonal conjugate of X(3570)
X(3572) = X(813)-Ceva conjugate of X(292)
X(3572) = X(665)-cross conjugate of X(649)
X(3572) = crosspoint of X(i) and X(j) for these (i,j): (292,813), (1438,2702)
X(3572) = crosssum of X(i) and X(j) for these (i,j): (239,812), (659,2238)
X(3572) = crossdifference of every pair of poins on the line X(238)X(239)
X(3572) = trilinear pole of line X(512)X(1015)

### X(3573) = TRILINEAR PRODUCT X(6)*X(3570)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 - bc)/(b - c)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3573) lies on these lines: 1,21   2,1083   99,110   100,101   109,932   192,2175   660,2284   666,885   692,1492   785,835   898,901   1332,1633   1618,2397

X(3573) = isogonal conjugate of X(876)
X(3573) = cevapoint of X(238) and X(659)
X(3573) = X(659)-cross conjugate of X(238)
X(3573) = crosspoint of X(99) and X(666)
X(3573) = crosssum of X(512) and X(665)
X(3573) = crossdifference of every pair of points on the line X(244)X(661)
X(3573) = intersection of tangents to Steiner circumellipse at X(99) and X(666)

### X(3574) = INTERSECTION OF LINES X(4)X(54) AND X(5)X(51)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[a2(b2 + c2) - (b2 - c2)2](2a6 + b6 + c6 - 3a4b2 - 3a4c2 - b2c4 - b4c2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3574) = X(21)-of-the-orthic-triangle if ABC is acute. Let A'B'C' be the orthic triangle, which is the pedal (and cevian) triangle of the orthocenter, H = X(4) . The Euler lines of the triangles HB'C', HC'A', HA'B' concur in X(3574). (Michel Garitte, July 7, 2009)

Let A'B'C' be the cevian triangle of X(5). Let La be the reflection of line B'C' in line BC, and define Lb, Lc cyclically. Let A" = Lb ∩ Lc, and define B", C" cyclically. The lines AA", BB", CC" concur in the isogonal conjugate of X(3574). (Randy Hutson, January 29, 2018)

Let A'B'C' be the orthic triangle. X(3574) is the radical center of the circles O(3,4) of triangles AB'C', BC'A', CA'B'. (Randy Hutson, July 31 2018)

X(3574) lies on these lines: 4,54   5,51   25,2917   113,137   125,389   130,132   155,195   185,427   193,576   235,1843   1204,3541   1839,2182   1907,2883   3519,3527

X(3574) = midpoint of X(4) and X(54)
X(3574) = reflection of X(1209) in X(5)
X(3574) = X(4)-Ceva conjugate of X(3575)
X(3574) = crosspoint of X(4) and X(5)
X(3574) = crosssum of X(3) and X(54)

### X(3575) = EULER LINE INTERCEPT OF THE LINE X(64)X(66)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[2a6 + b6 + c6 - 3a4b2 - 3a4c2 - b4c2 - b2c4]/(b2 + c2 - a2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

As a point on the Euler line, X(3575) has Shinagawa coefficients (F, -E - 3F).

Let KH denote the hyperbola discussed at X(1112). As noted in the paper cited at X(1112), KH is the X(4)-Ceva conjugate of the Euler line. Inversely, the Euler line is the X(4)-Ceva conjugate of KH. Since X(3574) lies on KH, its X(4)-Ceva conjugate, which is X(3575), lies on the Euler line. (Peter Moses, July 7, 2009)

X(3575) lies on these lines: 1,1892   2,3   53,571   64,66   128,135   185,1503   225,1852   317,1975   515,1829   516,1902   1179,1300   1192,1853   1452,1837   1828,2829   1862,1872

X(3575) = reflection of X(1885) in X(4)
X(3575) = X(4)-Ceva conjugate of X(3574)
X(3575) = X(65)-of-orthic-triangle if ABC is acute
X(3575) = Kosnita-to-orthic similarity image of X(5)
X(3575) = {X(4),X(24)}-harmonic conjugate of X(5)
X(3575) = intersection of tangents to Hatzipolakis-Lozada hyperbola at X(4) and X(52)
X(3575) = crosspoint, wrt orthic triangle, of X(4) and X(52)
X(3575) = Ehrmann-vertex-to-orthic similarity image of X(5)

### X(3576) = INTERSECTION OF LINES X(1)X(3) AND X(2)X(515)

Trilinears    (3a - b - c)(a - b + c)(a + b - c) - 4abc
Trilinears    3 cos A + cos B + cos C - 1 : : (Peter Moses, March 10, 2011)

Trilinears    r + 2 R cos A : :
X(3576) = X(1) + 2*X(3)

X(3576) occurs in Hyacinthos #18095 (Quang Tuan Bui, August 2, 2009). Following a note by Seiichi Kirikami [see X(165)], Peter Moses, October 19, 2010, found that if DEF is the pedal triangle of X(3576), then |FB| + |CE| = |DC| + |AF| = |EA| + |BD|.

Let Pa be the parabola with focus A and directrix BC, and define Pb and Pc cyclically. X(3576) is the centroid of the six points of tangency of lines from X(1) to Pa, Pb, and Pc. (Randy Hutson, December 2, 2017)

Let A'B'C' be the hexyl triangle. Let Ab = BC∩C'A', Ac = BC∩A'B', and define Bc, Ba, Ca, Cb cyclically. Then X(3576) is the centroid of AbAcBcBaCaCb. (Randy Hutson, December 2, 2017)

Let I be the incenter, X(1), and Ia, Ib, Ic be the excenters. Let Oa be the de Longchamps circle of triangle IbIcI, and define Ob and Oc cyclically. Then X(3576) is the radical center of Oa, Ob, Oc. (Randy Hutson, December 2, 2017)

If you have The Geometer's Sketchpad, you can view X(3576).

X(3576) lies on these lines: 1,3   2,515   4,1125   8,3523   9,48   10,631   11,3523   20,946   21,84   30,1699   58,602   77,102   78,947   106,1292   140,355   154,392   187,1572   198,374   200,956   223,1455   284,2257   376,516   380,1108   405,1490   519,3158   549,952   573,1449   580,1468   581,1193   595,601   936,958   950,3086   953,1308   960,1071   962,3522   963,2910   970,1051   991,995   1001,1012   1151,1702   1152,1703   1210,3486   1212,3207   1350,1386   1435,1870   1511,2948   1732,2364   1766,3247   1829,3515   1902,3516   2202,3100   2320,3306   2718,2742

X(3576) = midpoint of X(1) and X(165)
X(3576) = reflection of X(i) in X(j) for these (i,j): (40,165),(165,3)
X(3576) = isogonal conjugate of X(3577)
X(3576) = X(2320)-Ceva conjugate of X(1)
X(3576) = crosssum of X(2) and X(2093)
X(3576) = crossdifference of every pair points on the line X(650)X(1769)
X(3576) = X(2)-of-hexyl-triangle
X(3576) = anticomplement of X(10175)
X(3576) = X(2)-of-2nd-circumperp-triangle
X(3576) = X(381)-of-excentral-triangle
X(3576) = {X(1),X(3)}-harmonic conjugate of X(40)
X(3576) = homothetic center of hexyl triangle and medial triangle of 2nd circumperp triangle
X(3576) = homothetic center of Johnson triangle and cross-triangle of Aquila and anti-Aquila triangles
X(3576) = Thomson-isogonal conjugate of X(9)
X(3576) = orthocenter of cross-triangle of ABC and anti-Aquila triangle
X(3576) = insimilicenter of circumcircles of ABC and anti-Aquila triangle; the exsimilicenter is X(1)
X(3576) = trisector nearest X(3) of segment X(1)X(3)
X(3576) = endo-homothetic center of Ehrmann side-triangle and orthic triangle; the homothetic center is X(381)

### X(3577) = ISOGONAL CONJUGATE OF X(3576)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(3a - b - c)(a - b + c)(a + b - c) - 4abc]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3577) lies on these lines: 1,227   4,3340   7,515   8,908   9,374   21,40   30,3255   80,1537   314,322   943,1697   971,1159   1000,1512   1156,2800   1172,2331   1476,3333   1709,3065   2320,3306   3243,3254

X(3577) = isogonal conjugate of X(3576)
X(3577) = cevapoint of X(1) and X(2093)
X(3577) = X(2099)-cross conjugate of X(1)
X(3577) = trilinear pole of line X(650)X(1769)

### X(3578) = INTERSECTION OF LINES X(2)X(6) AND X(8)X(30)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(2a + b + c)(a2 - b2 - c2 - bc)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3578) lies on these lines: 2,6   8,30   319,3219   340,445   519,2292   540,1046   542,1281

X(3578) = X(75)-Ceva conjugate of X(1125)

### X(3579) = INTERSECTION OF LINES X(1)X(3) AND X(10)X(30)

Trilinears    2a3 - b3 - c3 + (a2 + bc)(b + c) - 2a(b2 + c2 + bc) : :
Trilinears    r - 3 R cos A : :
X(3579) = X(1) - 3*X(3)

X(3579) lies on these lines: 1,3   4,2355   5,516   8,376   10,30   12,1770   20,355   24,1902   31,582   37,2160   42,500   44,573   45,1766   71,2173   72,74   109,227   140,946   191,210   267,2941   378,1829   381,1698   392,404   498,1836   515,550   518,3098   548,952   549,1125   631,962   759,1293   896,3214   901,2687   944,3522   971,1158   1191,1480   1216,2807   1250,2306   1656,1699   1702,3312   1703,3311   1827,1872   3085,3474   3218,3555

X(3579) = midpoint of X(i) and X(j) for these (i,j): (3,40), (20,355)
X(3579) = reflection of X(i) in X(j) for these (i,j): (946,140), (1385,3)
X(3579) = X(3210)-Ceva conjugate of X(37)
X(3579) = X(5)-of-1st-circumperp-triangle
X(3579) = X(140)-of-excentral-triangle
X(3579) = centroid of excenters and X(40)

### X(3580) = INTERSECTION OF LINES X(2)X(6) AND X(30)X(74)

Trilinears    bc[b6 + c6 + (a4 - b2c2)(b2 + c2) - 2a2(b4 + c4) + 2a2b2c2]
Trilinears    (csc A)(1 + cos 2B + cos 2C) : :

Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the orthic axis. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A"B"C" be the reflection of A'B'C' in the orthic axis. The triangle A"B"C" is homothetic to ABC, and its centroid is X(3580); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)

X(3580) lies on these lines: 2,6   5,568   22,1899   23,1503   24,68   30,74   52,1594   110,468   113,403   125,511   140,567   146,1514   186,2931   278,2994   281,445   287,2373   297,525   340,687   451,3193   467,2052   542,1495   1154,2072   1352,1995   1368,2979

X(3580) = midpoint of X(i) and X(j) for these (i,j): (23,3448), (265,3581)
X(3580) = reflection of X(i) in X(j) for these (i,j): (110,468), (146,1514), (858,125)
X(3580) = isotomic conjugate of X(2986)
X(3580) = complement of X(323)
X(3580) = X(340)-Ceva conjugate of X(30)
X(3580) = cevapoint of X(6) and X(2931)
X(3580) = X(3003)-cross conjugate of X(403)
X(3580) = crosspoint of X(i) and X(j) for these (i,j): (2,94), (76,1494)
X(3580) = crosssum of X(i) and X(j) for these (i,j): (6,50), (32, 1495), (647,2088)
X(3580) = crossdifference of every pair of points on the line X(184)X(512)
X(3580) = crosspoint of X(6) and X(2931) wrt both the excentral and tangential triangles
X(3580) = pole wrt polar circle of trilinear polar of X(1300)
X(3580) = X(48)-isoconjugate (polar conjugate) of X(1300)

### X(3581) = INTERSECTION OF LINES X(3)X(6) AND X(30)X(74)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[2b4 + 2c4 - a4 - a2b2 - a2c2 - 4b2c2][(b2 + c2 - a2)2 - b2c2]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let A1B1C1 and A2B2C2 be the 1st and 2nd Ehrmann circumscribing triangles. Let A* be the crosssum of A1 and A2, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(3581).

X(3581) lies on these lines: 3,6   30,74   49,1658   185,2937   186,323   381,1531   399,1495   541,1533   1204,1657

X(3581) = reflection of X(i) in X(j) for these (i,j): (265,3580), (323,1511), (399,1495)
X(3581) = perspector of ABC and cross-triangle of ABC and 2nd isogonal triangle of X(4)

### X(3582) = INTERSECTION OF LINES X(1)X(2) AND X(11)X(30)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[a4 + b4 + c4 + a2(3bc - 2b2 - 2c2) - 2b2c2]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(3582) = R*X(1) - 2rX(2)

X(3582) lies on these lines: 1,2   11,30   12,547   35,496   56,381   57,1727   58,3615   80,1319   244,1725   376,1479   497,3524   542,1428   553,1776   599,613   946,3336   1478,3545   1519,1768   1656,3304   1749,3218   1837,3656   2964,3075   3303,3526

### X(3583) = INTERSECTION OF LINES X(1)X(4) AND X(11)X(30)

Trilinears    bc(b4 + c4 - a4 - 2b2c2 + a2bc)
X(3583) = (R/r)*X(1) - 6X(2) + 4X(3)

X(3583) lies on these lines: 1,4   5,35   11,30   12,546   20,499   55,381   56,382   79,942   80,517   115,1914   149,519   316,350   484,516   495,3058   1203,1834   1539,3028   1749,1776   1770,3336   1781,1839   2308,3017   2964,3073   3065,3218   3070,3299   3071,3301   3100,3153

X(3583) = reflection of X(i) in X(j) for these (i,j): (36,11), (484,1737)
X(3583) = crosspoint of X(79) and X(80)
X(3583) = crosssum of X(35) and X(36)
X(3583) = homothetic center of 2nd isogonal triangle of X(1) and the reflection of the Johnson triangle in X(4); see X(36)
X(3583) = homothetic center of Mandart-incircle triangle and (cross-triangle of ABC and 2nd isogonal triangle of X(1))
X(3583) = homothetic center of Ehrmann vertex-triangle and intangents triangle
X(3583) = homothetic center of Ehrmann mid-triangle and Mandart-incircle triangle
X(3583) = {X(1),X(4)}-harmonic conjugate of X(3585)

### X(3584) = INTERSECTION OF LINES X(1)X(2) AND X(12)X(30)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[a4 + b4 + c4 - a2(3bc + 2b2 + 2c2) - 2b2c2]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(3584) = R*X(1) + 2rX(2)

X(3584) lies on these lines: 1,2   5,3058   11,547   12,30   13,1250   36,495   37,1989   55,381   79,3579   226,484   376,1478   388,3524   542,2330   553,3336   599,611   756,1725   1479,3545   1656,3303   1749,3219   2964,3074   3304,3526

### X(3585) = INTERSECTION OF LINES X(1)X(4) AND X(12)X(30)

Trilinears    bc(b4 + c4 - a4 - 2b2c2 - a2bc)
X(3585) = (R/r)*X(1) + 6X(2) - 4X(3)

X(3585) lies on these lines: 1,4   5,36   10,191   11,546   12,30   13,2307   20,498   55,382   56,381   65,79   115,172   149,3244   316,1909   355,1836   377,1698   535,2975   993,2476   1254,1725   1539,3024   1781,1826   2964,3072   3070,3301   3071,3299

X(3585) = reflection of X(35) in X(12)
> X(3585) = homothetic center of 2nd isogonal triangle of X(1) and Johnson triangle; see X(36)
X(3585) = {X(1),X(4)}-harmonic conjugate of X(3583)
X(3585) = homothetic center of Ehrmann vertex-triangle and anti-tangential midarc triangle

### X(3586) = INTERSECTION OF LINES X(1)X(4) AND X(30)X(57)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(2b4 + 2c4 - 3a4 + a3b + a3c - ab3 - ac3 + a2b2 + a2c2 + abc(b + c) + 2a2bc - 4b2c2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3586) lies on these lines: 1,4   7,3543   9,80   10,452   11,3576   20,1210   30,57   40,1728   46,2955   90,191   165,1737   329,519   355,1697   380,1826   405,1376   496,1420   516,2093   517,1864   936,2478   938,3146   993,1005   1449,1901   1453,1834

X(3586) = reflection of X(i) in X(j) for these (i,j): (1,497), (1750,4)
X(3586) = insimilicenter of hexyl and 2nd Johnson-Yff circles; the exsimilicenter is X(9614)
X(3586) = {X(1), X(1479)}-harmonic conjugate of X(9614)

### X(3587) = INTERSECTION OF LINES X(1)X(3) AND X(9)X(30)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a6 - a4(3b + c)(b + 3c) + a2(3b4 + 3c4 + 2b2c2 + 8bc3 + 8b3c) - (b - c)4(b + c)2
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3587) lies on these lines: 1,3   4,3305   9,30   20,3219   63,376   84,550   515,3358   582,1453   1445,3488   3306,3524

### X(3588) = CENTER OF THE MYAKISHEV CONIC

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)[a4(b + c) - b4(c + a) - c4(a + b) + b2c2(b + c) + c2a2(a - c) + a2b2(a - b) - 2ab2c2]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

In 2008, Alexei Myakishev gave the following construction of a conic. Let CA and CB be points on the line AB satisfying |BCA| = |CB| and |ACB| = |CA| and arranged in this order: CA, B, A, and CB. Define points AB, CB, BC, AC cyclically. The six points CA, BA, AB, CB, BC, AC lie on a conic. Myakishev's proof is by Carnot's theorem, since

[c/(c+a)][(a+b)/b][a/(a+b)][(b+c)/c][b/(b+c)][(c+a)/a] = 1.

For details, in Russian, visit Item 4 at Geometry.ru,

Peter Moses found that X(3588) is the point of concurrence of three lines constructed from points numbered 1,3,4,6,8,9,10,11, as follows:

X(3588) = X(8)X(573)∩X(2269,3057), where

X(573) = X(3)X(6)∩X(4)X(9)
X(2269) = X(8)X(9)∩X(1)X(573)
X(3057) = X(1)X(3)∩X(10)X(11)

The six points Ca, Ba, Ab, Cb, Bc, Ac can be constructed from the extangents triangle as follows. Let A'B'C' be the extangents triangle. Then Ab = BC∩C'A', Ac = BC∩A'B', and Bc, Ba, Ca, Cb are defined cyclically. (Randy Hutson, January 29, 2018)

X(3588) lies on these lines: 8,573   37,1953   42,181   71,594   213,2347   1824,2354   2225,2264   3059,3198

### X(3589) = ISOGONAL CONJUGATE OF X(3108)

Trilinears    bc(2a2 + b2 + c2) : :
Barycentrics    2a2 + b2 + c2 : a2 + 2b2 + c2 : a2 + b2 + 2c2 : :
X(3589) = 3X(2) + X(6)

In 2003, Peter Moses gave a general construction as follows: Suppose that P = u : v : w (barycentrics). The centroids of the triangles BCP, CAP, ABP form a triangle homothetic to ABC, with ratio -1/3 and center

P' = 2u + v + w : u + 2v + w : u + v +2w.

In 2010, Seiichi Kirikami gave another construction for P': let

D = AP∩BC,        E = BP∩CA,        F = CP∩AB.

Then the Newton lines of the quadrilaterals PEAF, PFBD, PDCE concur in P'. (The Newton line of a quadrilateral is the line of the midpoints of the two diagonals of the quadrilateral.)

X(3589) is the Moses-Kirikami image of the symmedian point. In the following list (from P. Moses, Aug. 23, 2010), the appearance of I, J means that X(j) is the Moses-Kirikami image of X(i).

1,1125    2,6   3,140   4,5   6,3589   7,142   8,10   20,3   23,468   69,141   99,620   100,3035   144,9   145,1   146,113   147,114   148,115   149,11   150,116   151,117   152,118   153,119   192,37   193,6   194,39   239,3008   315,626   316,625   329,3452   376,549   381,547   382,546   385,230   390,1001   410,441   487,642   488,641   550,3530   616,618   617,619   621,623   622,624   627,629   628,630   631,632   633,635   634,636   637,639   638,640   944,1385   962,946   1278,75   1320,1387   1330,3454   1370,1368   1654,1213   1657,548   1916,2023   1992,597   2475,442   2888,1209   2895,1211   3091,1656   3146,4   3151,440   3153,2072   3164,216   3177,1212   3180,396   3181,395   3241,551   3434,2886   3436,1329   3448,125   3522,631   3523,3526   3529,550   3543,381

Early in 2012, Seiichi Kirikami found a simple relation between the tetrahedron and the Moses-Kirikami image. Using 3-dimensional cartesian coordinates, suppose that the vertices of triangle ABC are placed in the xy-plane:

A = (x1, y1, 0), B = (x2, y2, 0), C = (x3, y3, 0).

Let P = (x4, y4, z4) be an arbitrary point not in the plane of ABC. Then the centroid of the tetrahedron ABCP is the point

Q = ( (x1 + x2 + x3 + x4)/4, (y1 + y2 + y3 + y4)/4, z4/4 ).

The orthogonal projections of P and Q onto the xy plane coincide with the Moses-Kirikami images of X(6) and X(3589). That is, X(6) has coordinates (x4, y4, 0), and X(3589) has coordinates ( (x1 + x2 + x3 + x4)/4, (y1 + y2 + y3 + y4)/4, 0 ).

Let A' be the midpoint of segment AX(6), and define B' and C' cyclically. Then A'B'C' is homothetic to the medial triangle, and the homothetic center is X(3589). (Randy Hutson, December 26, 2015)

Let E be the bicevian conic of X(2) and X(6); i.e., the ellipse that passes through the vertices of the medial and symmedial triangles. Then X(3589) is the center of E. This ellipse is also the locus of centers of circumconics passing through X(6). Let L be a line through X(2), and let P and P' be the points of intersection of L and the circumcircle. Let V be the locus of the crosssum of P and P'. The locus of V generated by L is E. (Randy Hutson, December 26, 2015)

Let A' be the reflection of X(6) in line BC. Let Oa be the circle with center A' and tangent to BC. Define Ob and Oc cyclically. The radical center of Oa, Ob, Oc is X(3589). (Randy Hutson, December 26, 2015)

If you have The Geometer's Sketchpad, you can view X(3589).

X(3589) lies on these lines:

2,6   5,182   10,1386   11,2330   12,1428   23,2916   39,698   51,3313   53,458   83,316   140,143   216,441   239,594   264,1990   373,468   397,622   398,621   427,1974   498,613   499,611   518,1125   542,547   549,3098   575,3564   576,632   625,2030   631,1350   742,3008   894,1086   1351,3526   1352,1656   1506,1692   1698,3416   2450,3566

X(3589) = midpoint of X(i) and X(j) for these (i,j): (2,597), (5,182), (6,141), (10, 1386), (625, 2030)
X(3589) = isogonal conjugate of X(3108)
X(3589) = complement of X(141) = complement of complement of X(6)
X(3589) = {X(2),X(6)}-harmonic conjugate of X(141)
X(3589) = centroid of {A,B,C,X(6)}
X(3589) = centroid of PU(11)PU(45)
X(3589) = Kosnita(X(6),X(2)) point
X(3589) = crosspoint of X(2) and X(83)
X(3589) = X(620) of 1st Brocard triangle
X(3589) = antipode of X(141) in conic {{X(13),X(14),X(15),X(16),X(141)}}
X(3589) = complement of isotomic conjugate of cevapoint of X(2) and X(6)
X(3589) = polar conjugate of isogonal conjugate of X(22352)
X(3589) = perspector of the medial triangle and the tangential triangle, wrt the symmedial triangle, of the bicevian conic of X(2) and X(6)
X(3589) = crossdifference of every pair of points on line X(523)X(2076)
X(3589) = X(9)-of-submedial-triangle if ABC is acute

### X(3590) = 1st DIXIT POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc/(b2 + c2 - a2 + 6*area(ABC))
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = 1/(2 cos A + 3 sin A)
Barycentrics   1/(2SA + 3S) : 1/(2SB + 3S) : 1/(2SC + 3S)     [Conway notation]

On Dec. 22, 2000, Atul Dixit constructed this point in Hyacinthos message #2183, as follows. Let ABC be a triangle with medians AD, BE, FC and centroid G. Construct semicircles with diameters BD, DC, BC outwardly. Let TA be the circle tangent to the three semicircles BD, DC, BC, and let A' be the center of TA. Define B' and C' cyclically. Then the lines AA', BB', CC' concur in X(3590). Barycentric coordinates were found by Paul Yiu (2000), and further properties and related points, by Peter Moses (2010).

The radius of circle TA is |BC|/6, and the Kiepert angle (e.g., B-to-C-to-X(3890)) is arctan(2/3).

If you have The Geometer's Sketchpad, you can view X(3590) and Dixit Chains of Circles.

X(3590) lies on the Kiepert hyperbola and these lines: {2,3594}, {4,6221}, {6,3591}, {20,1327}, {76,3595}, {140,3316}, {485,3523}, {486,5056}, {590,1131}, {1132,3068}, {1271,5490}, {1328,3091}, {1656,3317}, {3543,6482}

### X(3591) = 2nd DIXIT POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc/(b2 + c2 - a2 - 6*area(ABC))
Barycentrics   1/(2SA - 3S) : 1/(2SB - 3S) : 1/(2SC - 3S)     [Conway notation]
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = 1/(2 cos A - 3 sin A)

This point is obtained in the manner of X(3590) using the inward semicircles instead of outward.

If you have The Geometer's Sketchpad, you can view X(3591).

X(3591) lies on the Kiepert hyperbola and these lines: 2,3592   6,3590   20,1328   76,3593   140,3317   486,3523   615,1132  1131,3069  1327,3091   1656,3316

### X(3592) = ISOGONAL CONJUGATE OF 1st DIXIT POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 - a2 + 6*area(ABC))
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = 2 cos A + 3 sin A
Barycentrics   a2(2SA + 3S) : b2(2SB + 3S) : c2(2SC + 3S)     [Conway notation]

X(3592) lies on these lines: 2,3591   3,6   485,546   487,3589   590,1588   615,3525   1587,3529   2066,3298   2067,3297  3068,3071  3070,3146

X(3592) = radical center of the Lucas(3) circles
X(3592) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,3594), (6,371,1151), (6,1151,1152), (61,62,3311), (371,3311,6)

### X(3593) = ISOTOMIC CONJUGATE OF 1st DIXIT POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b2 + c2 - a2 + 6*area(ABC))
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = (3 + 2 cot A) csc A
Barycentrics   2SA + 3S : 2SB + 3S : 2SC + 3S     [Conway notation]

X(3593) lies on these lines: 2,6   4,1165   76,3591   488,3091   637,3523   641,1588

X(3593) = {X(2),X(69)}-harmonic conjugate of X(3595)

### X(3594) = ISOGONAL CONJUGATE OF 2nd DIXIT POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 - a2 - 6*area(ABC))
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = 2 cos A - 3 sin A
Barycentrics   a2(2SA - 3S) : b2(2SB - 3S) : c2(2SC - 3S)     [Conway notation]

X(3594) lies on these lines: 2,3590   3,6   486,546   488,3589   590,3525   615,1587   1588,3529   3069,3070   3071,3146  3297,3303  3298,3304

X(3594) = radical center of the Lucas(-3) circles
X(3594) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,3592), (6,372,1152), (6,1152,1151), (61,62,3312), (372,3312,6)

### X(3595) = ISOTOMIC CONJUGATE OF 2nd DIXIT POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b2 + c2 - a2 - 6*area(ABC))
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = (3 - 2 cot A) csc A
Barycentrics   2SA - 3S : 2SB - 3S : 2SC - 3S     [Conway notation]

X(3595) lies on these lines: 2,6   4,1164   76,3590   487,3091   638,3523   642,1587

X(3595) = {X(2),X(69)}-harmonic conjugate of X(3593)

### X(3596) = 1st ODEHNAL POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b3c3(b + c - a)
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = csc2A csc2(A/2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let A', B', C' denote the respective excircles of a triangle ABC. Let A'' be the circle tangent to A', B', C' whose interior includes A', and likewise for circles B'' and C''. Let Kaa be the point of tangency of circles A' and A'', and likewise for Let Kbb and Let Kcc. The lines A-to-Kaa, B-to-Let Kbb, C-to-Let Kcc concur in X(3596). For more, see

Boris Odehnal, "Some Triangle Centers Associated with the Circles Tangent to the Excircles," http://forumgeom.fau.edu/FG2010volume10/FG201006.pdf

Barycentrics for the center of the circle A'' are given by Peter Moses (August 26, 2014):

-2 a^4 (b+c)-(b-c)^2 (b+c)^3+a (b^2-c^2)^2+a^2 (b+c) (b^2+c^2)-a^3 (3 b^2+4 b c+3 c^2)
: (b-c) c^2 (b+c)^2+a (b-c) c (b+c) (2 b+c)+a^2 (b+c) (2 b^2+c^2)+a^3 (2 b^2+2 b c+c^2)
: (a-b) b^2 (a+b)^2+(a-b) b (a+b) (2 a+b) c+(a+b) (2 a^2+b^2) c^2+(2 a^2+2 a b+b^2) c^3

If you have The Geometer's Sketchpad, you can view X(3596) and X(3597).

X(3596) lies on these lines: 2,1240   8,314   10,75   43,350   69,150   86,996   92,2064   190,573   192,2092   219,645   261,958   264,1969   281,345   304,309   305,561   312,2321   346,646   730,1740

X(3596) = isogonal conjugate of X(1397)
X(3596) = isotomic conjugate of X(56)
X(3596) = isotomic conjugate of isogonal conjugate of X(8)
X(3596) = isotomic conjugate of isogonal conjugate of anticomplement of X(1)
X(3596) = polar conjugate of X(608)
X(3596) = trilinear product of extraversions of X(7)
X(3596) = trilinear product of vertices of Gemini triangle 39

### X(3597) = 2nd ODEHNAL POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc/[a5 + a4(b + c) - a3(b - c)2 - a2(b + c)(b2 + c2) - 2abc(b2 + bc + c2) - 2b2c2(b + c)]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let MA, MB, MC be the centers of the circles A'', B'', C'' used to construct X(3596), respectively. Then the lines A-to-MA, B-to-MB, C-to-MC concur in X(3597). For more, see the reference and sketch at X(3596).

Let P and Q be the intersections of line BC and the excircles radical circle. Let X = X(10). Let A' be the circumcenter of triangle PQX, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(3597). (cf. X(592), where the circle is the 1st Lemoine circle and X = X(182)) (Randy Hutson, January 15, 2019)

X(3597) lies on these lines: 2,970   4,2092   226,986   429,2052

### X(3598) = 1st LIU POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(a - b + c)(a + b - c)(3a2 + b2 + c2 - 2bc)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (a - b + c)(a + b - c)(3a2 + b2 + c2 - 2bc)

Let OA be the circle passing through B and C, internally tangent to the incircle, and likewise for OB and OC. Let PA be the point where OA meets the incircle, and likewise for PB and PC. Let QA be the point of intersection of the tangents to the incircle at PB and PC, and likewise for PB and PC. Then X(3598) is the perspector of the triangles PAPBPC and QAQBQC; this point is also X(7)-of-QAQBQC. This point and X(3599) were discovered by Kang-Ying Liu of St. Andrew's Priory School, Honolulu, Hawaii, during 2010.

X(3598) lies on these lines: 2,7   22,347   25,1119   56,105   145,3212   269,479   354,3056   1002,1469   1418,3290   1420,3160   1429,2280

### X(3599) = 2nd LIU POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(a - b + c)(a + b - c)(5a4 + b4 + c4 - 8a3b - 8a3c - 4b3c - 4bc3 + 6b2c2 + 2a2b2 + 2a2c2 + 12a2bc)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let PAPBPC be as at X(3598). Let TA be the point of intersection of the lines BPC and CPB. X(3599) is the perspector of triangles PAPBPC and TATBTC.

X(3599) lies on these lines: 2,7   55,479   165,279   651,1190

### X(3600) = INTERSECTION OF LINES X(1)X(7) AND X(8)X(57)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(a - b + c)(a + b - c)(3a2 + b2 + c2 + 2bc)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2a - [(a+b+c)^2]/(b + c - a)
X(3600) = 4(R/r)*X(1) + 3X(2) - 4X(3)

X(3600) was found by Peter Moses in connection with the 1st Liu point.

X(3600) lies on these lines: 1,7   2,12   3,1056   4,496   8,57   10,3361   21,1617   30,1058   36,3085   55,3522   65,145   100,1466   144,960   171,1106   193,330   226,452   346,2285   354,3486   376,3295   391,1400   443,956   474,3421   495,631   497,3146   515,938   519,3339   553,3241   942,944   948,1104   982,1254   1010,1014   1043,1434   1159,1483   1319,3485   1385,3487   1435,1891   1446,3424   1478,3086   1479,3543   2646,3475   3057,3474

X(3600) = reflection of X(8) in X(1706)
X(3600) = reflection of X(938) in X(3333)
X(3600) = {X(1),X(20)}-harmonic conjugate of X(390)

### X(3601) = INTERSECTION OF LINES X(1)X(3) AND X(9)X(21)

Trilinears    3 cos A + cos B + cos C + 1 : :

X(3601), the {X(1), X(3611)}-harmonic conjugate of X(3576), was found by Peter Moses in connection with the 1st Liu point.

Let Ja, Jb, Jc be the excenters and I the incenter. Let A' be the centroid of JbJcI, and define B' and C' cyclically. A'B'C' is also the cross-triangle of the excentral and 2nd circumperp triangles. A'B'C' is homothetic to the intouch triangle at X(3601). (Randy Hutson, July 31 2018)

X(3601) lies on these lines: 1,3   2,950   5,3586   7,3522   8,3158   9,21   10,3486   20,226   28,33   37,610   58,212   73,991   77,738   84,943   100,1706   142,390   200,958   376,3487   386,1453   405,936   443,497   452,3452   515,3085   516,3485   579,1449   631,1210   728,2329   938,3523   1012,1490   1055,3100   1104,2999   1193,2293   1253,1468   1439,3532   1682,3056   1698,1837   1876,3516

X(3601) = {X(1),X(3)}-harmonic conjugate of X(57)
X(3601) = homothetic center of 2nd Johnson-Yff triangle and cross-triangle of Aquila and anti-Aquila triangles

Points Associated with Morley Cubics

Points X(3602)-X(3609) are associated with the three Morley Cubics indexed as K29, K30, K31 at Bernard Gibert's Cubics in the Triangle Plane. The points are related to those in the section just before X(3272), under the heading "Points Associated with Equilateral Triangles." See also

### X(3602) = ISOGONAL CONJUGATE OF X(3274)

Trilinears        csc(A/3 - π/3) : csc(B/3 - π/3) : csc(C/3 - π/3)
Trilinears        cos(A/3) - 2 cos(B/3) cos(C/3) : cos(B/3) - 2 cos(C/3) cos(A/3) : cos(C/3) - 2 cos(A/3) cos(B/3) (Bernard Gibert, January 10, 2013)
Barycentrics   (sin A)csc(A/3 - π/3) : (sin B)csc(B/3 - π/3) : (sin C)csc(C/3 - π/3)

X(3602) is {X(357), X(358)}-harmonic conjugate of X(356). (Bernard Gibert, November 3, 2010).

Let A'B'C' be the 2nd Morley triangle. Let Ba and Ca be points on BC such that A'BcCa is an equilateral triangle; define Cb and Ac cyclically, and define Ab and Bc cyclically. Let A'' be the midpoint of Ab and Ac, and define B'' and C'' cyclically. Then A''B''C'', here named the 2nd MORLEY-MIDPOINT TRIANGLE, is an equilateral triangle that is homothetic to A'B'C' and perspective to ABC, with perspector X(3602). (Thanh Oai Dao and Peter Moses, March 30, 2018)

If you have GeoGebra, you can view X(3602).

X(3602) lies on these lines: 356,357   1134,1137   1135,1136   3274, 3604   3275,3603

### X(3603) = ISOGONAL CONJUGATE OF X(3275)

Trilinears        csc(A/3 ) : csc(B/3) : csc(C/3)
Barycentrics   (sin A)csc(A/3) : (sin B)csc(B/3) : (sin C)csc(C/3)

X(3603) is {X(1136), X(1137)}-harmonic conjugate of X(3276). (Bernard Gibert, November 3, 2010).

Let A'B'C' be the 3rd Morley triangle. Let Ba and Ca be points on BC such that A'BcCa is an equilateral triangle; define Cb and Ac cyclically, and define Ab and Bc cyclically. Let A'' be the midpoint of Ab and Ac, and define B'' and C'' cyclically. Then A''B''C'', here named the 3rd MORLEY-MIDPOINT TRIANGLE, is an equilateral triangle that is homothetic to A'B'C' and perspective to ABC, with perspector X(3603). (Thanh Oai Dao and Peter Moses, March 30, 2018)

X(3603) lies on these lines: 357,1135   358,1134   1136,1137   3273, 3604   3275,3602

### X(3604) = ISOGONAL CONJUGATE OF X(3273)

Trilinears         csc(A/3 + π/3) : csc(B/3 + π/3) : csc(C/3 + π/3)
Barycentrics   (sin A)csc(A/3 + π/3) : (sin B)csc(B/3 + π/3) : (sin C)csc(C/3 + π/3)

X(3604) is {X(1134), X(1135)}-harmonic conjugate of X(3277). (Bernard Gibert, November 3, 2010)

Let A'B'C' be the 1st Morley triangle. Let Ba and Ca be points on BC such that A'BcCa is an equilateral triangle; define Cb and Ac cyclically, and define Ab and Bc cyclically. Let A'' be the midpoint of Ab and Ac, and define B'' and C'' cyclically. Then A''B''C'', here named the 1st MORLEY-MIDPOINT TRIANGLE, is an equilateral triangle that is homothetic to A'B'C' and perspective to ABC, with perspector X(3604). (Thanh Oai Dao and Peter Moses, March 30, 2018)

X(3604) lies on these lines: 356,1134   357,1137   358,1136   3273,3603   3274,3602

### X(3605) = ISOGONAL CONJUGATE OF X(356)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/[cos A/3 + 2 cos B/3 cos C/3]
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3605) lies on these lines: 358,3279   1507,1508

### X(3606) = ISOGONAL CONJUGATE OF X(3276)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/[cos(A/3 - 2π/3) + 2 cos(B/3 - 2π/3) cos(C/3 - 2π/3)]
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3606) lies on this line: 1137,3281

### X(3607) = ISOGONAL CONJUGATE OF X(3277)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/[cos(A/3 - 4π/3) + 2 cos(B/3 - 4π/3) cos(C/3 - 4π/3)]
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3607) lies on this line: 1135,3279

### X(3608) = 1st MORLEY-GIBERT PERSPECTOR

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos A + 2 cos B cos C + 3 cos(A/3) + 6 cos(B/3) cos(C/3)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3608) is the perspector of the Morley triangle and the triangle of the cusps of the Steiner deltoid. (Bernard Gibert, November 3, 2010). See

X(3608) lies on these lines: 5,356   20,3278   631,3279   3273,3609

### X(3609) = 2nd MORLEY-GIBERT PERSPECTOR

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos A + 2 cos B cos C + 3 sin(A/3+π/6) + 6 sin(B/3) sin(C/3)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(3609) is the perspector of the triangle of the cusps of the Steiner deltoid and the equilateral triangle J defined just before X(3272). (Bernard Gibert, November 3, 2010)

X(3609) lies on these lines: 5,3272   20,3334   631,3335

### X(3610) = 1st AYME-MOSES PERSPECTOR

Trilinears    bc(b + c)(a2 - b2 - c2)(a2 + b2 + c2 + 2bc) : :

In a Hyacinthos message dated January 10, 2011, Jean-Louis Ayme introduced a triangle as follows. Let RA be the radical axis of the circumcircle and the A-excircle, and define RB and RC cyclically. Let TA = RB∩RC, and define TB and TC cyclically. (TA is also the radical center of the circumcircle and the B- and C- excircles.) The Ayme triangle TATBTC is perspective to triangle ABC and also perspective to many other triangles. Peter Moses found that its perspector with the cevian triangle of X(346) is X(3610). He also found that the A-vertex of the Ayme triangle has first barycentric as follows:

- (b + c)(a2 + b2 + c2 + 2bc) : b(a2 + b2 - c2) : c(a2 - b2 + c2),

from which the other two vertices are easily obtained. The Ayme triangle is perspective to ABC with perspector X(19).

Moses found that the locus of X such that the cevian triangle of X is perspective to the Ayme triangle is a cubic which passes through the points X(i) for i = 1, 2, 19, 75, 279, 304, 346, 2184. A barycentric equation for this Ayme-Moses cubic follows:

(Cyclic sum of ayz[by(a2 + b2 - c2) - cz(a2 - b2 + c2] ) = 0.

The Ayme triangle is homothetic to the incentral triangle, and the center of homothety is X(612). (Randy Hutson, September 14, 2016)

X(3610) lies on these lines: 10,37   19,346   612,2345

X(3610) = perspector of ABC and cross-triangle of ABC and Ayme triangle

### X(3611) = 2nd AYME-MOSES PERSPECTOR

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c) (b2 + c2 - a2)[a3(b + c) + (b - c)2(b2 + c2 - a2 - ab - ac + 2bc)]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

In a Hyacinthos message dated January 7, 2011, Jean-Louis Ayme noted that the orthic triangle of ABC is perspective to the medial triangle of the extangents triangle of ABC. Peter Moses found coordinates for the perspector, X(3611).

X(3611) lies on these lines: 19,51   25,3197   40,185   42,1409   55,184   65,225   71,228   209,3198   511,3101   1899,2550

### X(3612) = INTERSECTION OF LINES X(1)X(3) AND X(21)X(90)

Trilinears    3 cos A + cos B + cos C : :

Let Ja, Jb, Jc be the excenters and I the incenter. Let A' be the centroid of JbJcI, and define B' and C' cyclically. A'B'C' is also the cross-triangle of the excentral and 2nd circumperp triangles. A'B'C' is homothetic to the reflection triangle of X(1) at X(3612). (Randy Hutson, July 31 2018)

X(3612) lies on these lines: 1,3   9,2174   21,90   78,993   80,1698   140,1837   186,1452   284,1723   376,1770   377,1125   442,3586   498,515   499,950   550,1836   631,1737   1006,1728   1047,3362   1449,2245   1788,3524   1905,3515   3474,3528

X(3612) = {X(1),X(3)}-harmonic conjugate of X(46)

### X(3613) = ISOTOMIC CONJUGATE OF X(1078)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc/(b2c2 + c2a2 + a2b2 - a4)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(3613) = X(311) + (1 - 2 cos(2ω))*X(325)

Let A' be the point of intersection of the tangents to the nine-point circle at the points where the circle meets line BC, and define B' and C' cyclically. Then the lines AA', BB', CC' concur in X(3613). Also, A'B'C' is the side-triangle of the tangential triangle of the medial triangle and the tangential triangle of the orthic triangle. (Randy Hutson, August 30, 2011.)

X(3613) is the pole of the Lemoine axis with respect to the nine-point circle. (Luis González, Hyacinthos #20253, October 5, 2011)

X(3613) is the perspector of the nine-point circle and lies on the hyperbola that passes through the points A, B, C, X(4), X(5). (Randy Hutson, December 30, 2012.)

X(3613) lies on these lines: 4,160   5,141   53,232   66,2548   157,3425   184,2980   311,325   1316,3447   1485,3148

X(3613) = isogonal conjugate of X(5012)
X(3613) = isotomic conjugate of X(1078)

X(3613) = X(8053)-of-orthic-triangle if ABC is acute

### X(3614) = 1st HUTSON-FEUERBACH POINT

Trilinears       1 + 3 cos(B - C) : 1 + 3 cos(C - A) : 1 + 3 cos(A - B)
= 1 - 3 cos2(B/2 - C/2) : 1 - 3 cos2(C/2 - A/2) : 1 - 3 cos2(A/2 - B/2)
Barycentrics   (sin A)(1 + 3 cos(B - C)) : (sin B)(1 + 3 cos(C - A)) : (sin C)(1 + 3 cos(A - B))
X(3614) = s*R*X(1) + 3*S*X(5)

Let A'B'C' be the Feuerbach triangle, L the line through A and X(5), and A'' = L∩B'C'; define B'' and C'' cyclically. Then the lines A'A'', B'B'', C'C'' concur in X(3614). X(3614) = {X(5),X(12)}-harmonic conjugate of X(11). Further, X(3614) is the trilinear pole (with respect to the Feuerbach triangle) of the perspectrix of ABC and the Feuerbach triangle. (Randy Hutson, August 30, 2011.)

X(3614) lies on these lines: 1,5   35,546   55,3091   56,3090   140,3585   381,498   442,1155   899,3136   1329,2476   1478,1656   1698,1836   2475,3035   2635,3142   3058,3085

### X(3615) = 2nd HUTSON-FEUERBACH POINT

Trilinears        1/(cos(B - A) + cos(C - A)) : 1/((cos(C - B) + cos(A - B)) : 1/(cos(A - C) + cos(B - C))
= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c - a)/[(b + c)(b2 + c2 - a2 + bc)] (Peter Moses, September 5, 2011)

Barycentrics   (sin(A))/(cos(B - A) + cos(C - A)) : (sin(B))/((cos(C - B) + cos(A - B)) : (sin(C))/(cos(A - C) + cos(B - C))

Let A'B'C' be the Feuerbach triangle, L the line through A' and X(5), and A'' = L∩BC; define B'' and C'' cyclically. Then the lines AA'', BB'', CC'' concur in X(3615). (Randy Hutson, August 30, 2011.)

X(3615) = isogonal conjugate of X(2594)

X(3615) lies on these lines: 1,564   2,582   5,49   11,60   12,59   21,36   29,1870   58,3582   86,1443   476,953   501,3583   655,2595   946,1325   2287,2323

Homothetic Centers: 3616 - 3637

Suppose that X is a triangle center, that M is the medial triangle, and that t is a real number. The t-dilation of M from X, denoted by H(X; M, t), is a triangle center. If X = x : y : z (trilinears), then

H(X; M, t) = bc[(1 + t)ax + (1 - t)by + (1 - t)cz] : ca[(1 - t)ax + (1 + t)by + (1 - t)cz] : ab[(1 - t)ax + (1 - t)by + (1 + t)cz],

with inverse given by

H-1(X; M, t) = bc[(1 - t)(by + cz) - 2ax] : ca[(1 - t)(cz + ax) - 2by : ab[(1 - t)(ax + by) - 2cz].

César E. Lozada contributed several such triangle centers (March 21, 2011), as summarized here:

TABLE 1: H(X; M, t)
X t=1/2 t=2 t=-1/2 t=-2
X(1) X(3616) X(145) X(667) X(3617)
X(3) X(631) X(20) X(3090) X(3091)
X(4) X(3091) X(3146) X(3523) X(3522)
X(5) X(1656) X(4) X(3526) X(631)
X(6) X(3618) X(193) X(3619) X(3620)
X(8) X(3617) X(3621) X(3622) X(3623)
X(10) X(1698) X(8) X(3624) X(3616)

TABLE 2: H-1(X; M, t)
X t=1/2 t=2 t=-1/2 t=-2
X(1) X(3244)X(1125) X(3625) X(3626)
X(3) X(550) X(140) X(3627) X(546)
X(4) X(382) X(5) X(1657) X(550)
X(5) X(546) X(3628) X(548) X(3530)
X(6) X(3630) X(3589) X(3631) X(3632)
X(8) X(3633) X(10) X(3634) X(3244)
X(10) X(3626) X(3635) X(3636) X(3637)

### X(3616) = H(X(1); M, 1/2)

Trilinears    bc(3a + b + c) : :
Trilinears    r + R sin B sin C : :
Barycentrics   3a + b + c : a + 3b + c : a + b + 3c
X(3616) = 2*X(1) + 3*X(2) = 3*X(8) - 8*X(10)

Let OA be the circle tangent to side BC at its midpoint and to the circumcircle on the side of BC opposite A. Define OB and OC cyclically. Let LA be the external tangent to circles OB and OC that is the nearer of the two to OA. Define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. Then A'B'C' is homothetic to ABC, and the center of homothety is X(3616). See the reference at X(1001).

X(3616) lies on these lines: 1,2   3,962   4,1385   5,944   7,21   9,1475   11,2476   12,1388   20,946   29,278   37,2272   40,3306   55,404   63,3333   69,1386   85,3160   100,474   104,3560   105,1036   106,835   140,1482   142,390   149,214   226,452   238,1468   244,986   320,3246   329,405   330,1655   346,3247   354,960   355,3090   377,497   388,1319   391,1449   392,942   406,1870   442,496   443,1058   459,1829   515,3091   516,3522   517,631   940,1191   941,2277   952,1656   958,3304   966,1100   968,988   982,2292   1000,1392   1056,3436   1220,2899   1320,3035   1376,3303   1479,2475   1699,3146   1788,2099   1962,3210   3218,3338   3242,3589   3524,3579

X(3616) = isogonal conjugate of X(2334)
X(3616) = {X(1),X(2)}-harmonic conjugate of X(8)
X(3616) = {X(1),X(10)}-harmonic conjugate of X(145)
X(3616) = {X(2),X(145)}-harmonic conjugate of X(10)
X(3616) = X(10)-of-cross-triangle-of-Aquila-and-anti-Aquila-triangles

### X(3617) = H(X(1); M, -2)

Trilinears    bc(a - 3b - 3c) : :
Trilinears    2 r - 3 R sin B sin C : :
Barycentrics    a - 3b - 3c : b - 3c - 3a : c - 3a - 3b
X(3617) = 4 X(1) - 9 X(2) = X(7) + 2 X(8) + 2 X(9) = X(8) + 4 X(10)

X(3617) lies on these lines: 1,2   20,355   40,3146   44,391   45,346   63,1706   88,1219   100,958   144,1654   149,1145   193,3416   321,341   377,3421   390,1837   404,956   442,1159   452,3419   496,1000   515,3522   517,3091   631,952   944,3523   984,1278   1376,2975   1482,3090   1483,3526   1697,3305   1788,3600   2551,3434

X(3617) = isotomic conjugate of X(30712)
X(3617) = insimilicenter of Spieker circle and AC-incircle
X(3617) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,8,145), (8,10,2)

### X(3618) = H(X(6); M, 1/2)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c)=bc(3a2 + b2 + c2)
Barycentrics   3a2 + b2 + c2 : 3b2 + c2 + a2 : 3c2 + a2 + b2

X(3618) lies on these lines: 1,344   2,6   4,83   8,1386   140,1351   159,1995   239,2345   264,1249   297,3087   316,2030   348,1445   388,1428   393,458   459,1843   487,3311   488,3312   497,2330   511,631   575,1352   576,3525   611,3086   613,3085   625,1692   1350,3523   1503,3091   1656,3564   2297,2999   3060,3313   3098,3524

### X(3619) = H(X(6); M, -1/2)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c)=bc(a2 + 3b2 + 3c2)
Barycentrics   a2 + 3b2 + 3c2 : b2 + 3c2 + 3a2 : c2 + 3a2 + 3b2

X(3619) lies on these lines: 2,6   4,3096   182,3525   511,3090   631,1352   1350,3091   1503,3523   3526,3564

### X(3620) = H(X(6); M, -2)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c)=bc(a2 - 3b2 - 3c2)
Barycentrics   a2 - 3b2 - 3c2 : b2 - 3c2 - 3a2 : c2 - 3a2 - 3b2

X(3620) lies on these lines: 2,6   8,1738   20,1352   66,3410   76,2996   145,3416   253,3164   320,2345   340,3087   511,3091   631,3564   1350,3146   1351,3090   1353,3526   1503,3522   1843,2979

### X(3621) = H(X(8); M, 2)

Trilinears    bc(5a - 3b - 3c) : :
Trilinears    4 r - 3 R sin B sin C : : : :
Barycentrics   5a - 3b - 3c : 5b - 3c - 3a : 5c - 3a - 3b
X(3621) = 8 X(1) - 9 X(2) = 5 X(8) - 4 X(10)

X(3621) lies on these lines: 1,2   20,952   44,346   45,391   63,2136   89,1219   149,3436   377,1159   517,3146   518,1278   631,1483   944,3522   1482,3091   1697,3219

### X(3622) = H(X(8); M,-1/2)

\ Trilinears    bc(5a + b + c) : :
Trilinears    2 r + R sin B sin C : :
Barycentrics    5a + b + c : 5b + c + a : 5c + a + b
X(3622) = 4 X(1) + 3 X(2)

X(3622) lies on these lines: 1,2   7,1420   20,1385   21,999   56,1621   79,2320   81,1191   86,3445   100,3303   144,1001   149,377   193,1386   320,1279   388,1388   390,2646   391,1100   404,3295   452,3487   4496,2476   497,2475   517,3523   631,1482   940,1616   944,3091   946,3146   952,3090   962,3522   1056,2478   1219,1255   1319,3485   1483,1656   1697,3306   2325,3247   3218,3333

X(3622) = {X(1),X(2)}-harmonic conjugate of X(145)

### X(3623) = H(X(8); M, -2)

Trilinears    bc(7a - b - c)
Trilinears    4 r - R sin B sin C : : : :
Barycentrics   7a - b - c : 7b - c - a : 7c - a - b
X(3623) = 8 X(1) - 3 X(2)

X(3623) lies on these lines: 1,2   4,1483   20,1483   20,1482   79,1392   100,3304   144,3243   149,388   193,3242   346,1100   390,2098   517,3522   944,3146   952,3091   1056,2475   1120,2334   1320,3296   1449,2325   1697,3218   2099,3600   2136,3306   2975,3303

### X(3624) = H(X(10); M, 1/2)

Trilinears   bc(3a +2 b + 2c) : :
Trilinears    bc + rR : ca + rR : ab + rR
Trilinears    r + 4 R sin B sin C : :
Barycentrics   3a + 2b + 2c : 3b + 2c + 2a : 3c + 2a + 2b
X(3624) = X(1) + 6 X(2) = 3 X(8) - 10 X(10)

Let Ja, Jb, Jc be the excenters and I the incenter. Let A' be the centroid of JbJcI, and define B' and C' cyclically. A'B'C' is also the cross-triangle of the excentral and 2nd circumperp triangles. A'B'C' is homothetic to the 3rd Euler triangle at X(3624). (Randy Hutson, July 31 2018)

X(3624) lies on these lines: 1,2   3,1699   5,3576   9,583   11,3601   12,1420   34,451   35,474   36,405   40,140   57,191   58,748   63,3337   90,3255   165,631   226,3361   377,3583   442,3586   443,1479   515,3090   516,3523   517,3526   595,750   940,1203   1213,1449   1385,1656   1706,3035   2478,3585   3306,3336   3339,3485

X(3624) = complement of X(667)
X(3624) = {X(1),X(2)}-harmonic conjugate of X(1698)
X(3624) = homothetic center of ABC and cross-triangle of Aquila and anti-Aquila triangles

### X(3625) = H-1(X(1); M, -1/2)

Trilinears    bc(4a - 3b - 3c) : :
Trilinears    7 r - 6 R sin B sin C : : : :
Barycentrics   4a - 3b - 3c : 4b - 3c - 3a : 4c - 3a - 3b
X(3625) = 7 X(1) - 9 X(2) = X(1) - 3 X(8) = 2 X(8) - X(10)

X(3625) lies on these lines: 1,2   44,2321   72,2802   515,1657   536,1358   548,952

X(3625) = {X(1),X(8)}-harmonic conjugate of X(3626)
X(3625) = {X(8),X(10)}-harmonic conjugate of X(4669)

### X(3626) = H-1(X(1); M, -2)

Trilinears    bc(2a - 3b - 3c) : :
Trilinears    5 r - 6 R sin B sin C : :
Barycentrics    2a - 3b - 3c : 2b - 3c - 3a : 2c - 3a - 3b
X(3626) = 5 X(1) - 9 X(2) = X(1) + 3 X(8) = X(1) - 3 X(10) = X(8) + X(10)

X(3626) lies on these lines: 1,2   40,3529   44,594   45,2321   106,1339   355,382   515,550   517,546   952,3530   960,2802

X(3626) = midpoint of X(8) and X(10)
X(3626) = {X(1),X(2)}-harmonic conjugate of X(15808)
X(3626) = {X(1),X(8)}-harmonic conjugate of X(3625)
X(3626) = {X(1),X(10)}-harmonic conjugate of X(3634)

### X(3627) = H-1(X(3); M,-1/2)

Trilinears    bc[a2(b2 + c2 - 4a2) + 3(b2 - c2)2] : :
X(3627) = X(3) - 3*X(4) = 3*X(381) + 5*X(382)

As a point on the Euler line, X(3627) has Shinagawa coefficients (1, -7).

X(3627) lies on these lines: 2,3   53,3284   143,185   156,1514   495,3585   496,3583   515,1483   576,1353   1173,3521   1478,3303   1479,3304   1484,2829   1493,2883   1533,3574

X(3627) = {X(3),X(4)}-harmonic conjugate of X(547)
X(3627) = X(1482)-of-orthic-triangle if ABC is acute
X(3627) = {X(381),X(382)}-harmonic conjugate of X(5073)
X(3627) = X(20)-of-Ehrmann-mid-triangle
X(3627) = Johnson-to-Ehrmann-mid similarity image of X(382)

### X(3628) = H-1(X(5); M, 2)

Trilinears    bc[a2(-5b2 - 5c2 + 2a2) + 3(b2 - c2)2] : :
Trilinears    5 cos A + 6 cos B cos C : :
Trilinears   6 sec A + 5 sec B sec C : :
X(3628) = 3*X(2) + X(5) = X(3) + 3*X(5)

As a point on the Euler line, X(3628) has Shinagawa coefficients (5, 1).

X(3628) is the centroid of the set {A', B', C', X(5)}, where A'B'C' is the medial triangle; more generally, H-1(X; M, 2) is the centroid of the set {A', B', C', X}. (Angel Montesdeoca, 12/20/2011)

X(3628) lies on these lines: 2,3   17,395   18,396   32,3054   39,3055   52,373   141,576   143,1216   156,182   230,1506   233,3284   485,3594   486,3592   495,499   496,498   575,3564   623,630   624,629   952,1125   1209,1493

X(3628) = centroid of ABCX(5)
X(3628) = complement of X(140)
X(3628) = Kosnita(X(5),X(2)) point
X(3628) = center of conic which is locus of centers of circumconics passing through X(5)
X(3628) = center of bicevian conic of X(2) and X(5)
X(3628) = {X(2),X(5)}-harmonic conjugate of X(140)
X(3628) = {X(3),X(5)}-harmonic conjugate of X(546)
X(3628) = homothetic center of X(2)-altimedial and X(140)-anti-altimedial triangles

### X(3629) = H-1(X(6); M, 1/2)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c)= bc(4a2 - b2 - c2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3629) lies on these lines: 2,6   53,648   182,3530   317,1990   382,1351   397,621   398,622   487,3594   488,3592   511,550   518,3244   542,1539   546,576   1112, 1843   1350,3528

### X(3630) = H-1(X(6); M, -1/2)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c)= bc(4a2 - 3b2 - 3c2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3630) lies on these lines: 2,6   53, 340   548,3098   1205,2854   1503,1657

### X(3631) = H-1(X(6); M, -2)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c)= bc(2a2 - 3b2 - 3c2)
Barycentrics   2a2 - 3b2 - 3c2 : 2b2 - 3c2 - 3a2 : 2c2 - 3a2 - 3b2

X(3631) lies on these lines: 2,6   319,1086   320,594   338,1232   382,1352   511,546   545,2321   550,1503   1350,3529   3530,3564

### X(3632) = H-1(X(8); M, 1/2)

Trilinears    bc(3a - 2b - 2c) : ca(3b - 2c - 2a) : ab(3c - 2a - 2b)
Trilinears    5 r - 4 R sin B sin C : :
Barycentrics    3a - 2b - 2c : 3b - 2c - 2a : 3c - 2a - 2b
X(3632) = 5 X(1) - 6 X(2) = 3 X(8) - 2 X(10)

X(3632) lies on these lines: {1,2}, {3,5288}, {4,4900}, {5,16200}, {6,4007}, {9,3943}, {35,956}, {36,5687}, {37,4034}, {40,550}, {44,4873}, {45,4727}, {46,6762}, {55,5258}, {56,17573}, {57,10944}, {58,4720}, {63,11010}, {65,3894}, {69,1266}, {72,3586}, {75,4888}, {80,3680}, {90,12641}, {100,7280}

X(3632) = {X(8),X(10)}-harmonic conjugate of X(4668)
X(3632) = homothetic center of outer Garcia triangle and mid-triangle of medial and anticomplementary triangles

### X(3633) = H-1(X(8); M, -1/2)

Trilinears    bc(5a - 2b - 2c) : :
Trilinears    7 r - 4 R sin B sin C : :
Barycentrics   5a - 2b - 2c : 5b - 2c - 2a : 5c - 2a - 2b
X(3633) = 7 X(1) - 6 X(2) = 5 X(8) - 6 X(10)

X(3633) lies on these lines: 1,2   40,548   46,2136   191,1697   210,1357   517,1657   1317,1420   1482,1699   1483,3576   1743,2325   2093,3189   3339,3476

### X(3634) = H-1(X(10); M, 2)

Trilinears    bc(2a + 3b + 3c) : :
Trilinears    r - 6 R sin B sin C : :
Barycentrics   2a + 3b + 3c : 2b + 3c + 3a : 2c + 3a + 3b
X3634) = X(1) + 3*X(10) = 3*X(2) + X(10) = X(1) - 9 X(2) = X(8) - 5 X(10)

X(3634) lies on these lines: 1,2   5,516   37,1574   40,3090   44,1213   46,3305   88,1224   140,515   165,3091   355,3526   442,1155   451,1861   474,993   632,1385   750,1724   944,3533   946,1656   1739,2292   3039,3161   3219,3336   3525,3576

X(3634) = {X(1),X(10)}-harmonic conjugate of X(3626)
X(3634) = {X(2),X(10)}-harmonic conjugate of X(1125)
X(3634) = centroid of ABCX(10)
X(3634) = complement of X(1125)
X(3634) = Kosnita(X(10),X(2)) point
X(3634) = homothetic center of medial triangle and midpoint triangle of X(10)
X(3634) = center of conic which is locus of centers of circumconics passing through X(10)
X(3634) = center of bicevian conic of X(2) and X(10)
X(3634) = {X(1),X(2)}-harmonic conjugate of X(19862)
X(3634) = {X(8),X(10)}-harmonic conjugate of X(4745)

### X(3635) = H-1(X(10); M, -1/2)

Trilinears    bc(6a - b - c) : :
Trilinears    7 r - 2 R sin B sin C : :
Barycentrics    6a - b - c : 6b - c - a : 6c - a - b
X(3635) = 7X(1) - 3X(2) = 7 X(1) - 3 X(2) = 3 X(1) - X(10) = 3 X(8) - 5 X(10)

Recalling that triangle centers are functions, at (a,b,c) = (6,9,13), the values of X(3635) and X(15519) are equal.

X(3635) lies on these lines: {1, 2}, {6, 4029}, {20, 11224}, {21, 13602}, {37, 4700}, {44, 4982}, {65, 3892}, {72, 3898}, {75, 4909}, {86, 4464}, {149, 5270}, {214, 13996}, {355, 5072}, {392, 4005}, {405, 8162}, {726, 4718}, {2796, 4409}, {2802, 9945}, {3555, 3962}, {3630, 5847}, {4072, 4898}

X(3635) = midpoint of X(i) and X(j) for these {i,j}: {1, 3244}, {10, 145}, {944, 4301}, {1482, 5882}, {2650, 4065}, {3057, 3874}, {3555, 3878}, {3625, 3633}, {3924, 5262}, {4084, 5697}, {4297, 7982}, {4314, 12559}, {5493, 11531}, {10912, 12437}, {12675, 13600}
X(3635) = reflection of X(i) in X(j) for these (i,j): (8, 3634), (10, 3636), (1125, 1), (3625, 4691), (3626, 1125), (3632, 4746), (3754, 5045), (4665, 4758), (4701, 10), (4745, 551), (6684, 15178), (7317, 15223), (13753, 5570)
X(3635) = complement of X(3625)
X(3635) = anticomplement of X(4691)
X(3635) = {X(1),X(10)}-harmonic conjugate of X(3636)

### X(3636) = H-1(X(10); M, -2)

Trilinears    bc(6a + b + c) : :
Trilinears    5 r + 2 R sin B sin C : :
Barycentrics   6a + b + c : 6b + c + a : 6c + a + b
X(3636) = 5 X(1) + 3 X(2) = 3 X(1) + X(10) = 3 X(8) - 7 X(10)

X(3636) lies on these lines: 1,2   86,1266   226,1388   382,946   515,546   516,550   517,3530   993,3304   3528,3576

X(3636) = {X(1),X(2)}-harmonic conjugate of X(3244)
X(3636) = {X(1),X(10)}-harmonic conjugate of X(3635)
X(3636) = midpoint of X(1) and X(1125) (the anticomplement and complement of X(10))

### X(3637) = ISOGONAL CONJUGATE OF X(3355)

Trilinears        1/x(3355) : 1/y(3355) : 1/z(3355)
Barycentrics   a/x(3355) : b/y(3355) : c/z(3355)

X(3637) = isogonal conjugate of X(3355)

X(3637) lies on the Darboux cubic and these lines: 4,3356   20,3355   1490,3472   1498,2130   3182,3353

### X(3638) = INNER SODDY-GERGONNE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[(a+b-c)(a-b+c)(a+b+c)(2a2-b2-c2-ab-ac+2bc) - 4*sqrt(3)*(2a3-b3-c3-a2b-a2c+b2c+bc2)(area(ABC)]

Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c)=(a+b-c)(a-b+c)(a+b+c)(2a2-b2-c2-ab-ac+2bc) + 4*sqrt(3)*(2a3-b3-c3-a2b-a2c+b2c+bc2)(area(ABC)

In the plane of triangle ABC, let X,Y,Z be the points of contact of the incircle with sidelines BC,CA,AB, and let X',Y',Z' be the harmonic conjugates of X,Y,Z with respect to {B,C}, {C,A}, {A,B}, respectively. It is well known that AX, BY, CZ concur in the Gergonne point, X(7), and that X',Y',Z' lie on the Gergonne line. The circles with centers X',Y',Z' passing respectively through X,Y,Z are orthogonal to the incircle and to the Gergonne line, forming a coaxal system with the Gergonne line as line of centers. The radical axis is the Soddy line, which passes through X(1) and X(7) and is the line of centers of the orthogonal coaxal system, which includes the Soddy circles. Among the points on this line are the two limiting points (point-circles of the system): X(3638) and X(3639). The former lies between X(7) and the inner Soddy point, X(175). (Richard Guy, September 2, 2011)

Coordinates and properties for the points P=X(3638) and Q=X(3639) were found by Peter Moses (September 3, 2011):
midpoint(P,Q) = X(1323)
crossdifference(P,Q) = X(657)
crosssum(P,Q) = X(36)
cevapoint(P,Q) = X(80)
idealpoint(P,Q) = X(516)
Q = {X(1),X(7)}-harmonic conjugate of P
Q = inverse-in-incircle of P

Write I = X(1), U = X(7), r = inradius, R = circumradius, and d = 4*sqrt(3)*area(ABC). Then distance ratios are given by |IP|/|PU| = 2r(r + 4R)/d and |IQ|/|QU| = -2r(r + 4R)/d

X(3638) lies on these lines: 1,7   14,226   57,1277   553,1082

### X(3639) = OUTER SODDY-GERGONNE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[(a+b-c)(a-b+c)(a+b+c)(2a2-b2-c2-ab-ac+2bc) + 4*sqrt(3)*(2a3-b3-c3-a2b-a2c+b2c+bc2)(area(ABC)]

Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c)=(a+b-c)(a-b+c)(a+b+c)(2a2-b2-c2-ab-ac+2bc) + 4*sqrt(3)*(2a3-b3-c3-a2b-a2c+b2c+bc2)(area(ABC)

See X(3638) for a description of the inner and outer points.

X(3639) lies on these lines: 1,7   13,226   57,1276   553,559

### X(3640) = 1st KIRIKAMI POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b2 + c2 - ab - ac + 2*area(ABC)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Seiichi Kirikami (November 1, 2010), defined a point P having actual trilinear distances (p,q,r) by the equations a + p = b + q = c + r and a point P' with distances (p',q',r') satisfying p - a = q - b = r - c. Here, P = X(3640) and P' = X(3641). Kirikami also discovered a bicentric pair of points satisfying p + b = q + c = r + a and another bicentric pair satisfying p - b = q - c = r - a. The six point are given by homogeneous trilinears t(a,b,c) : t(b,c,a) : t(c,a,b) as follows:

X(3640): t(a,b,c) = b2 + c2 - ab - ac + 2*area(ABC)
X(3641): t(a,b,c) = b2 + c2 - ab - ac - 2*area(ABC)
K1: t(a,b,c) = ac - b2 + 2*area(ABC)
K2: t(a,b,c) = ab - c2 + 2*area(ABC)
K3: t(a,b,c) = ac - b2 + 2*area(ABC)
K4: t(a,b,c) = ab -cb2 + 2*area(ABC)

Peter Moses (November 1, 2010) found that the six points lie on an ellipse, E, here called the Kirikami-Moses ellipse. The center of E is the incenter of ABC, and an equation for E is as follows: e(a,b,c,x,y,z) + e(b,c,a,y,z,x) + e(c,a,b,z,x,y) = 0, where

e(a,b,c,x,y,z) = h(a,b,c)x2 - j(a,b,c)yz, where

h(a,b,c) = (5a4 + b4 + c4 - 4a3b - 4a3c - 4a2bc + 2b2c2 + 2c2a2 + 2a2b2)

and

j(a,b,c) = a4 + b4 + c4 - 8b3c - 8bc3 + 6b2c2 - 2a2b2 - 2a2c2 + 8abc(b+c-a).

Moses found six more points on E:

M1 = (b - c) (a + b + c) + 2*sqrt(3)*area(ABC) : :
M2 = (b - c) (a + b + c) - 2*sqrt(3)*area(ABC) : :
M3 = 2ab + ac - b2 - 2c2 + 2*sqrt(3)*area(ABC)   :   :
M4 = 2ac + ab - c2 - 2b2 + 2*sqrt(3)*area(ABC)   :   :
M5 = 2ab + ac - b2 - 2c2 - 2*sqrt(3)*area(ABC)   :   :
M6 = 2ac + ab - c2 - 2b2 - 2*sqrt(3)*area(ABC)   :   :

These six points comprise three bicentric pairs: {M1, M2}, {M3, M4}, {M5, M6}.

Further properties found by Moses:

(1) The incenter is the midpoint of the segments X(3640)X(3641), K1K2, and K3K4.

(2) X(3640) and X(3641) lie on the line X(1)X(6).

(3) The lines K1K4 and K2K3 are parallel to the line X(1)X(6).

(4) The lines X(3640)K4, K1K3, and X(3641)K2 are parallel; indeed, the reflection of the first in the second is the third.

(5) The lines X(3640)K3, K2K3, and X(3641)K1 are parallel, and the reflection of the first in the second is the third.

(6) The lines tangent to E at X(3640) and X(3641) are perpendicular to the line X(1)X(3), as are the lines K1K2 and K3K4.

(7) The following four lines are parallel: the tangents to E at K2 and K4, and X(3640)K1 and X(3641)K3.

(8) The following four lines are parallel: the tangents to E at K1 and K3, and X(3640)K2 and X(3641)K4.

X(3640) lies on these lines: 1,6   8,175   481,2550   517,1160

### X(3641) = 2nd KIRIKAMI POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b2 + c2 - ab - ac - 2*area(ABC))

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

See X(3640) for a discussion of X(3641).

X(3641) lies on these lines: 1,6   8,176   482,2550   517,1161

### X(3642) = FERMAT POINT OF 1st BROCARD TRIANGLE

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(a6 - 2b6 - 2c6 - a4b2 - a4c2 + 2a2b4 + 2a2c4 + 2b4c2 + 2b2c4 + 6a2b2c2 + H), where
H = 4*sqrt(3)(a4 + 2b2c2)*area(ABC)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3642) and X(3644) lie on the line through X(141) parallel to the Euler line.

X(3642) lies on these lines: 2,14   3,618   4,636   13,76   16,298   17,628   30,141   32,395   62,633   69,532   381,624   530,599   616,2896   629,631   630,1656   2926,3131

### X(3643) = X(14) OF 1st BROCARD TRIANGLE

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(a6 - 2b6 - 2c6 - a4b2 - a4c2 + 2a2b4 + 2a2c4 + 2b4c2 + 2b2c4 + 6a2b2c2 - H), where
H = 4*sqrt(3)(a4 + 2b2c2)*area(ABC)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3643) lies on these lines: 2,13   3,619   4,635   6,532   14,76   15,299   18,627   30,141   32,396   61,634   69,533   381,623   531,599   617,2896   629,1656   630,631   2925,3132

### X(3644) = REFLECTION OF X(75) IN X(192)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(3/a - 2/b - 2/c)
Barycentrics   3/a - 2/b - 2/c : 3/b - 2/c - 2/a : 3/c - 2/a - 2/b
X(3644) = 9*X(2) - 10*X(37) = 6*X(2) - 5*X(75) = 3*X(2) - 5*X(192), etc.

For any point P, let L(A) be the line through P parallel to BC. Let U = AB∩L(A) and V = AC∩L(A), and let a' = |UV|. Define b' and c' cyclically. Let L be the line consisting of points P such that a + ta' = b + tb' = c + tc'. Then X(3644) is given by t=1. The points corresponding to t = -1, -1/2, and 0, are X(75), X(37), and X(192), respectively. (Seiichi Kirikami, November 7, 2010)

X(3644) = reflection of X(i) in X(j) for these (i,j): (75,192), (1278,37)

X(3644) lies on these lines: 2,37   190,1743   726,3244

### X(3645) = POINT ALKAID

Trilinears        f(A,B,C) : f(B,C,A) ; f(C,A,B), where f(A,B,C) = 1 + cos(B/2) + cos(C/2) - cos(A/2)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) ; (sin C)f(C,A,B)

Denote the incenter and excenters by I, IA, IB, IC. Let JA be the incenter of triangle BCI, and define JB and JC cyclically. The lines IAJA, IBJB, ICJC concur in X(3645).

X(3645) lies on the Euler line of the BCI triangle. (Randy Hutson, January 29, 2018)

X(3645) lies on these lines: 1,168   40,483   258,1127

### X(3646) = POINT ALMACH

Trilinears        f(A,B,C) : f(B,C,A) ; f(C,A,B), where f(A,B,C) = 5 + cos A + (3 + cos A)(csc A)(sin B + sin C)
= 1 + 2(1 + cos2(A/2)) cos(B/2) cos(C/2) cos(A/2)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) ; (sin C)f(C,A,B)
X(3646) = r*X(1) - 12R*X(2) - 2r*X(3)

Denote the incenter and excenters by I, IA, IB, IC. Let KA be the centroid of triangle BCI, and define KB and KC cyclically. The lines IAKA, IBKB, ICKC concur in X(3646).

Of the 2 intersections of the Bevan circle and line BC, let Ab be the one closer to B, and define Bc and Ca cyclically. Let Ac be the one closer to C, and define Ba and Cb cyclically. Let Ab' = {B,C}-harmonic conjugate of Ab, and define Bc' and Ca' cyclically. Let Ac' = {B,C}-harmonic conjugate of Ac, and define Ba' and Cb' cyclically. The points Ab', Ac', Bc', Ba', Ca', Cb' lie on an ellipse centered at X(3646). (Randy Hutson, December 10, 2016)

X(3646) lies on these lines: 1,210   2,40   3,2951   9,1125   10,1058   11,1697   57,191   405,1490   748,1453   936,1001   978,1045

Kirikami-Schiffler Points: 3647 - 3652

Suppose that X is a point and A'B'C' is a central triangle. Let LA be the line through A' parallel to the Euler line of triangle BCX, let LB be the line through B' parallel to the Euler line of CXA, and let LC be the line through C' parallel to the Euler line of AXB.

It is well known that if X=X(1), the incenter, then the three aforementioned Euler lines concur in the Schiffler point, X(21). If their parallels, the lines LA, LB, LC concur, the point of concurrence is the Kirikami-Schiffler point of the triangle A'B'C', denoted by KS(A'B'C'). Seiichi Kirikami (February 1, 2011) found that those lines concur if A'B'C' is the reference triangle ABC and also concur if A'B'C' is the medial triangle. Peter Moses found additional cases and properties. A summary follows:

triangle A'B'C' LA∩LB∩LC
ABC X(79)
medial X(3647)
excentral X(191)
anticomplementary X(3648)
intouch X(3649)
extouch X(3650)
Feuerbach X(442)
Fuhrmann X(191)
1st circumperp X(3651)
2nd circumperp X(21)
Carnot X(3652)

Suppose that A'B'C' is the cevian triangle of a point P. Then LA, LB, LC concur if and only if P lies on the cubic K455.

Suppose that A'B'C' is the anticevian triangle of a point P. Then LA, LB, LC concur if and only if P lies on the cubic given by the barycentric equation

a(2a + b + c)yz(y - z) + b(2b + c + a)zx(z - x) + c(2c + a + b)xy(x - y) = 0.

Suppose that A'B'C' is the pedal triangle of a point P on the line X(1)X(3). Then LA, LB, LC concur.

Suppose that A'B'C' is the antipedal triangle of a point P on the line X(1)X(4). Then LA, LB, LC concur.

### X(3647) = KS(MEDIAL TRIANGLE)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (2a + b + c)(b2 + c2 - a2 + bc)
= g(A,B,C) : g(B, C, A) : g(C, A, B), where (csc A) (2 sin A + sin B + sin C)(1 + 2 cos A)
Barycentrics   h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = (2 sin A + sin B + sin C)(1 + 2 cos A)
X(3647) = (3/2)X(2) - (1/2)X(79) = (3/2)X(21) - (1/2)X(1)

X(3647) lies on these lines: 1,21   2,79   9,2173   10,30   35,3219   44,2092   45,2305   100,3065   124,128   214,960   442,1155   540,3178   553,1125

### X(3648) = KS(ANTICOMPLEMENTARYTRIANGLE)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[(2 + 3R/r)S2 - (ac + 2SB)(ab + 2SC)]
= g(A,B,C) : g(B, C, A) : g(C, A, B), where g(A,B,C) = (csc A)[(2/R + 3/r)*area(ABC) - (sin A)(1 + 2 cos B)(1 + 2 cos C)]     (f and g from Wimalasiri Perera, 11/16/2011)
Trilinears        h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = bc[3a4 + 2a3(b + c) - a2(2b2 + 2c2 + bc) - 2a(b3 + c3) - abc(b + c) - (b2 - c2)2]     (Seiichi Kirikami, 12/13/2011)
X(3648) = 3X(2) - 2X(79) = (r + 4R)*X(7) - 2(r + 3R)*X(21)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3648) = anticomplement of X(79)

X(3648) lies on these lines: 2,79   7,21   8,30   10,191   40,153   63,2894   145,758   149,3065

### X(3649) = KS(INTOUCH TRIANGLE)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c)(2a + b + c)(a - b + c)(a + b - c)
= g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b + c)(2a + b + c)(a - b + c)(a + b - c)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Two proofs of Dao's theorem about cyclic hexagons were published in 2014: Nikolaos Dergiades's Dao's Theorem on Six Circumcenters Associated with a Cyclic Hexagon and Telv Cohl's A Purely Syntheitc Proof of Dao's Theorem on Six Circumcenters Associated with a Cyclic Hexagon. In the degenerate case that the hexagon is the intouch triangle A'B'C', the triangle of the circumcenters of AB'C', A'BC', A'B'C is perspective to A'B'C' with perspector X(3649). (Dao Thanh Oai, Francisco Javier Garcia Capitan, ADGEOM #1718, September 16, 2014)

X(3649) lies on these lines: 1,30   7,21   10,12   11,113   55,3487   57,191   140,3336   145,388   354,946   396,2306   429,1835   553,1125   940,1406   962,3303   1056,2098   1086,1193   1100,1839   1317,1365   1358,2795   1360,1367   1411,2647   1749,3337   1834,2650   1852,1870   1901,2294   3318,3324

### X(3650) = KS(EXTOUCH TRIANGLE)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(2a + b + c)(-2a3 + b3 + c3 - a2b - a2c - b2c - bc2 + 2ab2 + 2ac2 + 2abc)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(3650) lies on these lines: 8,30   9,46   21,999   553,1125   758,3057

### X(3651) = KS(1st CIRCUMPERP TRIANGLE)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = abcph + a2h2 - p(a - b)(a - c)[2abc + (b + c)(c + a - b)(a + b - c)], where p = a + b + c, h = b2 + c2 - a2
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(3651) = 3R*X(2) - (6R + 2r)*X(3)
X(3651) = 3X(3) - 2X(5428)
X(3651) = 3X(21) - 4X(5428)
X(3651) = 3X(165) - X(191)

As a point on the Euler line, X(3651) has Shinagawa coefficients (2\$a\$E + 2abc, - 3\$a\$E - 2abc).

X(3651) lies on these lines:
{2,3}, {4,442}, {9,2173}, {35,79}, {36,950}, {40,758}, {55,3487}, {56,3488}, {72,74}, {78,3587}, {81,500}, {98,1292}, {104,4297}, {108,1294}, {165,191}, {201,3465}, {212,1745}, {329,3648}, {477,1290}, {515,5258}, {581,1754}, {601,1742}, {842,2691}, {944,3428}, {1030,1901}, {1058,1617}, {1260,3650}, {1330,1792}, {1444,2893}, {1612,3772}, {1768,2949}, {1936,4303}, {2635,3074}, {2693,2766}, {2975,3419}, {3072,4300}, {3652,5777}, {4333,5010}, {5204,5427}, {5217,5714}, {5584,5657}, {5752,5890}

X(3651) = midpoint of X(2) and X(2475)
X(3651) = reflection of X(i) in X(j) for these (i,j): (4,442), (21,3)
X(3651) = X(78)-gimel conjugate of X(3430)
X(3651) = reflection of X(21) in X(3)
X(3651) = X(195)-of-2nd-extouch-triangle
X(3651) = X(i)-zayin conjugate of X(j) for these (i,j): (9,71), (522,6003), (3465,6000)

### X(3652) = KS(CARNOT TRIANGLE)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a6 + a4(bc - 3b2 - 3c2) + a3bc(b + c) + 3a2(b4 + c4) - abc(b + c)(b - c)2 - (b2 - c2)2(b2 + bc + c2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(3652) = (2r + 3R)*X(1) - 9R*X(2) - (4r - 3R)*X(3)

Quim Castellsaguer defines the Carnot triangle as the triangle formed by the circumcenters of the triangles BCH, CAH, ABH, where H denotes the orthocenter. The Carnot triangle is also known as the Johnson triangle.

X(3652) lies on these lines: 5,79   12,1727   21,104   ,30,40   140,1768   500,846   758,1482   942,1776   952,3065   3219,3579

### X(3653) = CENTROID OF {X(1), X(2), X(3)}

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(7a4 + b4 + c4 - 8a2b2 - 8a2c2 - 2b2c2 + 3k), where k = a(b3 + c3 - a2b - a2c - b2c - bc2 + 2abc)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(3653) = X(1) + X(2) + X(3)

X(3653) lies on these lines: 1,549   2,355   3,551   30,1699   381,1125   517,3524   631,3241   946,3534   3058,3612

### X(3654) = X(1)-EXTRAVERSION OF {X(1), X(2), X(3)}

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[a4 + b4 + c4 - 2(b2c2 + c2a2 + a2b2 - 3k)], where k = a(b3 + c3 - a2b - a2c - b2c - bc2 + 2abc)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(3654) = - X(1) + X(2) + X(3)

X(3654) lies on these lines: 1,549   2,392   3,519   8,376   10,381   30,40   63,1145   165,952   495,2093   515,3534   542,3416   547,1698   551,1482   962,3545   1385,3241   1836,3245

### X(3655) = X(2)-EXTRAVERSION OF {X(1), X(2), X(3)}

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(5a4 - b4 - c4 - 4a2b2 - 4a2c2 + 2b2c2 + 3k), where k = a(b3 + c3 - a2b - a2c - b2c - bc2 + 2abc)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(3655) = X(1) - X(2) + X(3)

X(3655) lies on these lines: 1,30   2,355   3,519   8,3524   40,1483   145,3579   376,517   381,515   549,952   1387,3586   1482,3534   1837,3582

### X(3656) = X(3)-EXTRAVERSION OF {X(1), X(2), X(3)}

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[a4 + b4 + c4 - 2(b2c2 + c2a2 + a2b2 + 3k)], where k = a(b3 + c3 - a2b - a2c - b2c - bc2 + 2abc)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(3656) = X(1) + X(2) - X(3)

X(3656) lies on these lines: 1,30   2,392   3,551   4,1392   8,3545   40,549   57,1387   119,3577   355,381   376,962   496,3340   516,3534   553,999   940,1480   944,3543   952,1699   1388,1770   3524,3579

### X(3657) = AYME PERSPECTOR

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 - c2)/[2a(b + c)sBsC - S2 - SBSC]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let A'B'C' be the medial triangle of the reference triangle ABC. Let P be the point of intersection of the lines X(1)X(3) and BC, and let P' be the point where the line through P perpendicular to line AX(3) meets that line. Let LA be the line PP', and define LB and LC cyclically. Let A''=LB∩LC, and define B'' and C'' cyclically. The lines AA'', BB'', CC'' concur in X(3657). (Jean-Louis Ayme, Hyacinthos #16676, August 21, 2008)

The Ayme triangle A''B''C'' is also perspective to these triangles: tangential, orthic, intangents, extangents, and the circumorthic. (Peter Moses, November 7, 2011)

X(3657) lies on these lines: 3,513   65,924   68,521   69,693   71,661   72,523   74,915   895,2990

### X(3658) = ISOGONAL CONJUGATE OF AYME PERSPECTOR

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [2a(b + c)sBsC - S2 - SBSC]/(b2 - c2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

As a point on the Euler line, X(3658) has Shinagawa coefficients (\$aSA3\$(E+F) -\$aSA2\$[(E+F)2-2S2] +3\$aSA\$FS2-\$a\$(E+F)FS2, 2\$aSBSC\$S2 -2\$aSA2\$S2).

X(3658) lies on the Euler line.

X(3658) lies on these lines: 2,3   100,110   108,925   109,2617   476,1290   1292,1302

### X(3659) = FEUERBACH POINT OF EXCENTRAL TRIANGLE

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos(A/2))/(sin(B/2) - sin(C/2))
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Let I be the incenter and IA the A-excenter of the triangle ABC. Let LA be the line joining the circumcenter of triangle BCI and the incenter of triangle BCIA, and define LB and LC cyclically. The lines LA, LB, LC concur in X(3659). (Seiichi Kirikami, April 12, 2010)

Let Ea be the ellipse with B and C as foci and passing through X(1), and define Eb and Ec cyclically. Let La be the line tangent to Ea at X(1), and define Lb and Lc cyclically. Let A' = La∩BC, B' = Lb∩CA, C' = Lc∩AB. Then A', B', C' are collinear, and the line A'B'C' meets the line at infinity at the isogonal conjugate of X(3659). (Randy Hutson, April 9, 2016)

X(3659) lies on the circumcircle and these lines: 3,164   55,258   106,1130   759,1128

X(3659) = X(119)-of-hexyl-triangle
X(3659) = center of hyperbola {X(1),X(164),X(166),X(167),X(168),excenters}
X(3659) = X(100) of 1st circumperp triangle
X(3659) = isogonal conjugate of infinite point of antiorthic axis of intouch triangle (or excentral triangle)

### X(3660) = MIDPOINT OF {X(3513), X(3514}

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = r2(r + 4R) + (r - 2R)sBsC
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a - b + c)(a + b - c)(b3 + c3 + a2b + a2c - 2ab2 - 2ac2 + 2abc - b2c - bc2)
X(3660) = (r2 + 2rR -2R2)*X(1) - r(r - 2R)*X(3)

Let CA be the circumcircle analog of the Conway circle; that is, the circle with center X(3) and radius (R2 + s2)1/2. Then X(3660) is the radical trace of CA and the Conway circle. (Randy Hutson, October 13, 2015)

If you have The Geometer's Sketchpad, you can view X(3660) as the radical trace of the incircle and circumcircle.

X(3660) lies on these lines: 1,3   11,971   105,2720   108,840   109,1279   222,614   244,1458   513,676   518,3035   1071,3086   1357,3322   1360,3025   1404,2246   1421,1456   1426,1878   1477,2222   1788,3555

X(3660) = inverse-in-incircle of X(57)
X(3660) = X(468)-of-intouch-triangle
X(3660) = inverse-in-{circumcircle, nine-point circle}-inverter of X(56)
X(3660) = radical trace of the incircle and circumcircle; see X(3513)
X(3660) = radical trace of circumcircle and Moses-Longuet-Higgins circle
X(3660) = radical trace of incircle and Moses-Longuet-Higgins circle
X(3660) = inverse-in-circumcircle of X(1617)

### X(3661) = 1st KIRIKAMI-MOSES POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b2 + c2 + bc)
Barycentrics   b2 + c2 + bc : c2 + a2 + ca : a2 + b2 + ab
X(3661) = 3(a2 + b2 + c2)*X(2) + (a + b + c)2X(8)
X(3661) = (a + b + c)2X(1) - 3(a2 + b2 + c2 + bc + ca + ab)X(2)
X(3661) = (bc + ca + ab)*X(75) + 2(a2 + b2 + c2)X(141)

X(3661) and X(3662) were considered by Seiichi Kirikami and Peter Moses in December, 2011, in connection with combos.

X(3661) lies on these lines: 1,2   6,319   7,3620   9,1654   63,2896   69,894   75,141   76,321   100,761   192,2321   226,3212   257,312   297,318   320,599   344,966   469,1829   1726,3219   2082,3305

X(3661) = isotomic conjugate of X(14621)

### X(3662) = 2nd KIRIKAMI-MOSES POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b2 + c2 - bc)
Barycentrics   b2 + c2 - bc : c2 + a2 - ca : a2 + b2 - ab

X(3662) = 3(a2 + b2 + c2)*X(2) + (a2 + b2 + c2 -2bc - 2ca - 2ab)X(7)
X(3662) = (a2 + b2 + c2 - bc - ca - ab)X(7) + (a2 + b2 + c2)*X(9
X(3662) = (bc + ca + ab)*X(75) - 2(a2 + b2 + c2)X(141)

X(3662) lies on these lines: 1,2896   2,7   6,320   8,1738   69,239   75,141   85,257   86,1333   273,297   306,3210   319,599   614,1716   982,2887   1266,1278   2345,3619

X(3662) = anticomplement of X(17353)

Combos Using Central Triangles

The discussion of combos near the beginning of ETC is continued here. Suppose that T is a central triangle, and let nT its normalization, so that the triangle nT is essentially a 3x3 matrix with row sums equal to 1, and the rows of nT are normalized barycentrics for the A-, B-, C- vertices of T.

Let X be a triangle center, given by barycentrics x : y : z, not necessarily normalized. The point whose rows are the matrix product X*(nT) is then a triangle center, denoted by Xcom(T).

Among central triangles T are cevian and anticevian triangles and others described at MathWorld. A brief list follows, with A-vertices given in barycentrics (not normalized):

Intouch triangle = cevian triangle of X(7)
A-vertex = 0 : 1/(c + a - b) : 1/(a + b - c)

Extouch triangle = cevian triangle of X(8)
A-vertex = 0 : c + a - b : a + b - c

Incentral triangle = cevian triangle of X(1)
A-vertex = 0 : b : c

Excentral triangle = anticevian triangle of X(1)
A-vertex = -a : b : c

Hexyl triangle
A-vertex = a(1 + a1 + b1 + c1) : b(-1 + a1 + b1 - c1) : c(1 + a1 - b1 + c1), where a1 = cos A, b1 = cos B, c1 = cos C

Tangential triangle = anticevian triangle of X(6)
A-vertex = -a2 : b2 : c2

1st Brocard triangle
A-vertex = a2 : c2 : b2

### X(3663) = X(6)com(INTOUCH TRIANGLE)

Barycentrics    (b - c)2 + a(b + c) : :

Let A'B'C' be the inverse-in-excircles triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(3663). (Randy Hutson, July 20, 2016)

X(3663) lies on these lines: 1,7   2,2415   6,527   9,3008   10,75   37,142   57,1766   63,1723   69,519   86,99   141,536   144,1743   165,3598   226,1465   256,2481   273,1785   307,1210   319,3625   320,3244   329,2999   545,3589   553,940   572,1429   573,1423   988,1125   1074,1111   1122,3057   1439,2823

X(3663) = isotomic conjugate of X(1222)
X(3663) = trilinear product of vertices of inverse-in-excircles triangle
X(3663) = perspector of ABC and cross-triangle of ABC and inverse-in-excircles triangle
X(3663) = complement of X(3729)
X(3663) = complement, wrt intouch triangle, of X(12723)

### X(3664) = X(37)com(INTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2a2 + a(b + c) - (b - c)2

X(3664) lies on these lines: 1,7   2,1743   6,142   10,69   11,1366   36,1014   37,527   57,573   58,86   75,519   222,226   255,307   319,3626   354,1122   511,942   948,1419   1086,1100   1099,1111   1266,3636   1394,3485   1396,1848   1565,2792

X(3664) = complement of X(4416)

### X(3665) = X(41)com(INTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 + c2)/(b + c - a)

X(3665) lies on these lines: 1,1565   5,1111   7,21   12,85   57,1759   65,760   226,241   269,1038   279,388   1086,2275   1355,1367   1441,3264   3160,3476

### X(3666) = X(42)com(INTOUCH TRIANGLE)

Trilinears        as + SA : bs + SB : cs + SC
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 + ab + ac)

Let A' be the trilinear pole of the tangent to the Apollonius circle where it meets the A-excircle, and define B' and C' cyclically. Triangle A'B'C' is homothetic to the medial triangle at X(3666). (Randy Hutson, January 15, 2019)

X(3666) lies on these lines: 1,3   2,37   6,63   7,941   9,2999,   11,114   21,1104   27,1841   31,1386   38,42   39,712   43,210   44,3219   45,3305   72,386   77,1407   81,593   88,1255   141,306   191,1203   226,1465   227,388   238,846   239,257   240,1859   244,1962   392,995   474,975   581,1071   612,1376   614,968   650,824   726,1215   756,899   896,2308   918,3310   960,1193   1015,2482   1036,3556   1054,1961   1150,3187   1211,2092   1279,1621   1355,1364   1396,1870   1762,2264   1817,2303   2236,2309

X(3666) = isogonal conjugate of X(2298)
X(3666) = isotomic conjugate of X(30710)
X(3666) = complement of X(321)
X(3666) = polar conjugate of isogonal conjugate of X(22345)
X(3666) = X(1215)com(Incentral) = X(1064)com[n(Inverse Hexyl)]

### X(3667) = X(513)com(INTOUCH TRIANGLE)

Barycentrics    (b - c)(3a - b - c) : :
X(3667) = X(513)com(Hexyl) = X(513)com(Excentral)

X(3667) lies on these lines:
4,2457   30,511   74,2758   98,2712   99,2705   100,2743   101,2737   102,2757   103,2726   104,2718   109,2731   110,2692   145,2403   572,1919   649,3239   885,3062   1027,1721   1292,2748   1294,2755   1295,2756   1296,2759   1297,2760   1519,1769   1768,2957   2394,3429   2487,2496

X(3667) = isogonal conjugate of X(1293)
X(3667) = excentral-isogonal conjugate of X(1054)
X(3667) = Thomson-isogonal conjugate of X(106)
X(3667) = Lucas-isogonal conjugate of X(106)

### X(3668) = X(219)com(INTOUCH TRIANGLE)

Trilinears    (cos B + cos C)/(1 + cos A) : :
Barycentrics    (b + c)/(b + c - a)2 : :

Let A'B'C' be the 3rd extouch triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(3668). (Randy Hutson, September 14, 2016)

Let A13B13C13 be Gemini triangle 13. Let A' be the center of conic {{A,B,C,B13,C13}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(3668). (Randy Hutson, January 15, 2019)

X(3668) lies on these lines: {1, 7}, {9, 948}, {10, 307}, {19, 57}, {37, 226}, {56, 2218}, {65, 1439}, {75, 1088}, {142, 241}, {158, 273}, {219, 527}, {222, 2219}, {342, 1785}, {387, 3339}, {534, 553}, {658, 897}, {759, 934}, {1020, 1400}, {1086, 1108}, {1122, 1358}, {1365, 2652}, {1367, 3324}, {1434, 2363}, {1445, 1723}, {1461, 1910}, {1486, 1617}, {2003, 2982}, {2217, 2385}

X(3668) = isogonal conjugate of X(2328)
X(3668) = isotomic conjugate of X(1043)
X(3668) = pole wrt polar circle of trilinear polar of X(2322)
X(3668) = X(48)-isoconjugate (polar conjugate) of X(2322)

### X(3669) = X(663)com(INTOUCH TRIANGLE)

Barycentrics   a(b - c)/(b + c - a) : :

X(3669) lies on these lines: {1, 3309}, {34, 2424}, {56, 667}, {57, 1022}, {65, 876}, {109, 1308}, {241, 514}, {278, 2401}, {513, 663}, {651, 3257}, {919, 934}, {1015, 1358}, {1019, 1429}, {1407, 2423}, {2526, 2530}, {4394, 4498}

X(3669) = isogonal conjugate of X(644)
X(3669) = isotomic conjugate of X(646)
X(3669) = complement of X(4462)
X(3669) = pole wrt incircle of line X(7)X(8)
X(3669) = trilinear pole of line X(244)X(1357)

### X(3670) = X(3293)com(INTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b3 + c3 + ab2 + ac2)

X(3670) lies on these lines: {1, 3}, {6, 1759}, {10, 38}, {58, 3218}, {63, 1724}, {72, 3216}, {79, 256}, {81, 849}, {191, 238}, {226, 1393}, {240, 1838}, {244, 1125}, {307, 1210}, {442, 1086}, {518, 3293}, {726, 1089}, {758, 1193}, {975, 3306}, {984, 1698}, {1046, 1203}, {1100, 3285}, {1107, 3125}, {3290, 3294}

### X(3671) = X(958)com(INTOUCH TRIANGLE)

Barycentrics   (b + c)(3a + b + c)/(b + c - a) : :

Let O* be a circle with center X(2) and variable radius R*. Let LA be the radical axis of O* and the A-excircle, and define LB and LC cyclically. Let A'=LB∩LC, and define B' and C' cyclically. There is a unique value of R* for which triangle A'B'C' is perspective to ABC, and the perspector is X(3671). (Randy Hutson, January 15, 2019)

X(3671) lies on these lines: {1, 7}, {2, 3339}, {10, 12}, {40, 3487}, {56, 551}, {57, 1125}, {142, 960}, {355, 1159}, {388, 519}, {496, 942}, {527, 958}, {938, 1699}, {950, 1836}, {1400, 3294}, {1697, 3475}, {1727, 3338}, {2093, 3085}, {2099, 3244}, {3361, 3616}, {3474, 3601}

### X(3672) = X(1449)com(INTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2 + b2 + c2 + 2ab + 2ca - 2bc

X(3672) lies on these lines: {1, 7}, {2, 37}, {6, 144}, {55, 3598}, {63, 2257}, {69, 145}, {81, 2255}, {86, 3445}, {142, 3247}, {190, 3618}, {239, 391}, {307, 938}, {319, 3621}, {320, 3623}, {527, 1449}, {941, 2481}, {999, 1014}, {1266, 3616}, {1423, 2269}, {1429, 2268}

X(3672) = isotomic conjugate of X(1219)
X(3672) = complement of X(4461)
X(3672) = anticomplement of X(2345)

### X(3673) = X(3501)com(INTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(a2 + b2 + c2 - 2bc)

X(3673) lies on these lines: {1, 85}, {2, 277}, {3, 1447}, {4, 7}, {10, 75}, {20, 3598}, {77, 1067}, {150, 1837}, {158, 331}, {169, 673}, {274, 988}, {304, 350}, {312, 1930}, {315, 320}, {318, 1235}, {348, 3086}, {496, 1565}, {517, 3212}, {1434, 3338}}

### X(3674) = X(2329)com(INTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 + c2 + ab + ac)/(b + c - a)

X(3674) lies on these lines: {1, 7}, {10, 3212}, {57, 348}, {65, 760}, {76, 85}, {142, 3061}, {274, 1432}, {527, 2329}, {552, 553}, {942, 1565}, {1125, 1447}, {1358, 2795}, {1441, 1930}, {1446, 2051}, {3598, 3616}

### X(3675) = X(1026)com(INTOUCH TRIANGLE)

Barycentrics    a(b - c)2(b2 + c2 - ab - ac) : :

X(3675) lies on these lines: {1, 3}, {11, 1111}, {63, 1083}, {244, 665}, {518, 1026}, {672, 1642}, {764, 1647}, {1027, 2424}, {1357, 2821}, {1463, 1736}, {2310, 3020}, {3218, 3573}

X(3675) = isogonal conjugate of X(5377)
X(3675) = crossdifference of every pair of points on line X(100)X(650)

### X(3676) = X(650)com(INTOUCH TRIANGLE)

Barycentrics   (b - c)/(b + c - a) : :

Let P1 and P2 be the two points on the Gergonne line whose trilinear polars are parallel to the Gergonne line. P1 and P2 lie on the circumconic centered at X(1086) (hyperbola {{A, B, C, X(2), X(7)}}), and circle {{X(2), X(109), X(675)}}. The midpoint of P1 and P2 is X(1638). X(3676) is the barycentric product P1*P2. (Randy Hutson, January 15, 2019)

X(3676) lies on these lines: {2, 4468}, {7, 6006}, {11, 3323}, {57, 649}, {109, 658}, {241, 514}, {278, 3064}, {513, 676}, {522, 693}, {918, 3239}, {934, 2222}, {1443, 1447}, {3321, 3322}}

X(3676) = isogonal conjugate of X(3939)
X(3676) = isotomic conjugate of X(3699)
X(3676) = complement of X(4468)
X(3676) = anticomplement of X(4521)
X(3676) = X(647)-of-intouch-triangle
X(3676) = trilinear pole of line X(1086)X(1358)

### X(3677) = X(2999)com(INTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 + 3b2 + 3c2 - 2bc)

X(3677) lies on these lines: {1, 3}, {9, 38}, {42, 3243}, {200, 3242}, {244, 612}, {518, 2999}, {613, 2003}, {1401, 3056}, {1449, 3509}, {2191, 2983}

### X(3678) = X(5)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(- a2 + b2 + c2 + bc)
X(3678) = X(550)com[Inverse(n(Hexyl))]

X(3678) lies on these lines: 1,748   3,2801   8,80   9,943   10,12   35,3219   38,3216   58,1757   78,993   100,191   200,1005   214,2975   386,872   392,3244   502,594   517,546   518,1125   519,960   537,596   551,3555   668,1237   740,3159   762,2295   765,1098   936,1445   942,3635   956,1388   976,1724   997,1420   2292,3293   3057,3625

X(3678) = X(1125)-of-inner-Garcia triangle

### X(3679) = X(3576)com(EXTOUCH TRIANGLE)

Trilinears    3 r - 4 R sin B sin C : :
Barycentrics    a - 2b - 2c : :
X(3679) = X(1) - 2*X(2) = X(8) + 2*X(10)

X(3679) lies on these lines: {1, 2}, {5, 3657}, {9, 80}, {12, 3340}, {30, 40}, {35, 958}, {36, 956}, {46, 529}, {63, 484}, {75, 537}, {100, 993}, {140, 3654}, {148, 1654}, {165, 376}, {210, 381}, {333, 3550}, {341, 1089}, {388, 553}, {516, 3543}, {518, 599}, {524, 3416}, {527, 1478}, {536, 984}, {540, 1046}, {542, 2948}, {544, 1282}, {549, 952}, {730, 3097}, {752, 1757}, {944, 3524}, {946, 3545}, {966, 2321}, {982, 1739}, {996, 1150}, {1213, 3247}, {1377, 3299}, {1378, 3301}, {1479, 2551}, {1573, 2276}, {1574, 2275}, {1697, 1837}, {1743, 2345}, {1768, 3359}, {1788, 3361}, {3208, 3294}, {3434, 3583}, {3436, 3585}, {3534, 3579}

X(3679) = midpoint of X(2) and X(8)
X(3679) = reflection of X(1) in X(2)
X(3679) = isogonal conjugate of X(2163)
X(3679) = complement of X(3241)
X(3679) = anticomplement of X(551)
X(3679) = {X(8),X(10)}-harmonic conjugate of X(1)
X(3679) = harmonic center of incircle and Spieker circle
X(3679) = harmonic center of Conway circle and excircles radical circle
X(3679) = X(165)com[Inverse(n(Hexyl))]
X(3679) = homothetic center of Caelum triangle and cross-triangle of Aquila and anti-Aquila triangles
X(3679) = outer-Garcia-to-ABC similarity image of X(2)
X(3679) = reflection of X(2) in X(10)
X(3679) = homothetic center of Gemini triangle 20 and cross-triangle of Gemini triangles 20 and 28

### X(3680) = X(2136)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a - b - c)/(3a - b - c)

Let A'B'C' be Triangle T(-1,3) (aka excenters-reflections triangle). Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(3680). (Randy Hutson, March 21, 2019)

Let A' be the orthocenter of BCX(1), and define B' and C' cyclically. A'B'C' is also the anticevian triangle, wrt intouch triangle, of X(1), and also the Garcia reflection triangle (Gemini triangle 8). X(3680) is the orthocenter of A'B'C'. (see also X(1699)) (Randy Hutson, March 21, 2019)

X(3680) lies on the Feuerbach hyperbola and these lines: {1, 474}, {4, 519}, {7, 145}, {8, 3452}, {9, 3057}, {10, 1000}, {21, 1697}, {40, 104}, {57, 1476}, {78, 1320}, {84, 517}, {200, 2098}, {518, 3062}, {728, 2170}, {941, 3169}, {1449, 2298}, {1482, 3577}, {2099, 2900}, {2320, 3601}, {3189, 3244}

X(3680) = isogonal conjugate of X(1420)
X(3680) = X(4)-of-Garcia-reflection-triangle

### X(3681) = X(1699)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 - ab - ac + bc)
X(3681) = X(3058)com(Excentral)

X(3681) lies on these lines: {1, 748}, {2, 210}, {4, 8}, {9, 1174}, {31, 1757}, {38, 43}, {42, 984}, {55, 1776}, {63, 100}, {65, 3617}, {69, 3263}, {78, 947}, {81, 612}, {144, 3059}, {145, 960}, {150, 2890}, {374, 391}, {390, 1864}, {392, 3241}, {511, 3578}, {561, 668}, {674, 3060}, {846, 2177}, {896, 3550}, {899, 982}, {986, 3214}, {1005, 2900}, {1376, 3218}, {1474, 2287}, {2836, 2895}, {3057, 3621}, {3555, 3616}

X(3681) = complement of X(4430)
X(3681) = anticomplement of X(354)
X(3681) = centroid of triangle T(a, c - b)
X(3681) = centroid of Gemini triangle 30

### X(3682) =  X(1715)com(EXTOUCH TRIANGLE)

Trilinears    (cos A) (cot A) (b + c) : :
Barycentrics    a2(b + c)(- a2 + b2 + c2)2 : :

X(3682) lies on these lines: {1, 2}, {3, 48}, {9, 581}, {20, 2947}, {35, 1819}, {56, 209}, {58, 2327}, {72, 73}, {97, 2169}, {100, 1816}, {101, 1297}, {210, 2594}, {226, 3191}, {238, 1612}, {255, 394}, {276, 313}, {283, 1794}, {326, 1264}, {329, 1745}, {518, 1066}, {580, 2323}, {603, 3173}, {758, 1042}, {901, 2755}, {908, 1838}, {960, 1064}, {1073, 1260}, {1217, 1826}, {1332, 1792}, {1472, 1918}, {1490, 2324}, {1780, 1801}, {2654, 3419}, {2939, 3101}

X(3682) = isogonal conjugate of X(8747)
X(3682) = crossdifference of every pair of points on line X(649)X(7649)

### X(3683) = X(3219)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(-a + b + c)(2a + b + c)

X(3683) lies on these lines: {2, 1155}, {3, 1709}, {6, 968}, {9, 55}, {21, 60}, {31, 37}, {38, 1279}, {42, 44}, {45, 612}, {63, 354}, {65, 405}, {144, 3475}, {191, 942}, {212, 1212}, {238, 846}, {381, 1698}, {392, 993}, {430, 1213}, {452, 1837}, {518, 1621}, {553, 1125}, {756, 902}, {940, 1707}, {958, 3057}, {1011, 2182}, {1100, 1962}, {1104, 2292}, {1214, 1456}, {1376, 3305}, {2187, 2267}, {2223, 3294}, {2293, 2318}, {2328, 2361}

### X(3684) = X(3509)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(-a + b + c)(a2 - b2)

X(3684) lies on these lines: {1, 2271}, {2, 2280}, {6, 43}, {8, 41}, {9, 55}, {72, 3496}, {78, 2082}, {100, 672}, {101, 519}, {193, 1958}, {218, 3501}, {219, 3169}, {220, 3208}, {238, 1914}, {239, 385}, {242, 740}, {261, 284}, {294, 2340}, {346, 3217}, {350, 3570}, {391, 2268}, {404, 1475}, {511, 3033}, {518, 910}, {521, 650}, {940, 1449}, {1743, 3550}, {1802, 3189}, {2266, 2550}, {2269, 2287}

### X(3685) = X(1757)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (-a + b + c)(a2 - bc)

X(3685) lies on these lines: {1, 87}, {2, 968}, {8, 9}, {20, 2128}, {21, 261}, {31, 1999}, {55, 312}, {75, 1001}, {100, 2726}, {105, 3263}, {149, 3006}, {190, 518}, {238, 239}, {242, 862}, {321, 1621}, {344, 2550}, {345, 497}, {350, 1281}, {519, 1757}, {522, 663}, {536, 1279}, {595, 2901}, {614, 3210}, {643, 2361}, {664, 1456}, {874, 1921}, {960, 1043}, {1045, 1050}, {1265, 3189}, {1266, 3616}

### X(3686) = X(37)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (-a + b + c)(2a + b + c)

X(3686) lies on these lines: {1, 966}, {2, 1449}, {6, 10}, {8, 9}, {37, 519}, {44, 594}, {45, 3625}, {69, 142}, {72, 2262}, {75, 527}, {141, 3008}, {145, 3247}, {198, 956}, {239, 1654}, {261, 284}, {306, 2280}, {515, 573}, {553, 3578}, {604, 1150}, {965, 1210}, {997, 3554}, {1100, 1125}, {1107, 2092}, {1743, 2345}, {2238, 2300}, {2287, 2323}, {2324, 2654}

X(3686) = complement of X(3879)

### X(3687) = X(171)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (-a + b + c)(b2 + c2 + ab + ac)

X(3687) lies on these lines: {1, 2}, {9, 345}, {57, 69}, {63, 573}, {72, 970}, {75, 226}, {181, 518}, {261, 284}, {312, 2321}, {318, 469}, {319, 2985}, {320, 553}, {321, 908}, {355, 2050}, {440, 2968}, {914, 1150}, {950, 1043}, {960, 1682}, {1211, 2092}, {1376, 1460}, {1738, 2887}, {1812, 2323}, {2895, 3218}

### X(3688) = X(75)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b + c - a)(b2 + c2)

X(3688) lies on these lines: {8, 314}, {9, 3056}, {31, 2273}, {37, 674}, {38, 1401}, {39, 1964}, {42, 2300}, {51, 756}, {55, 219}, {71, 2223}, {77, 1362}, {78, 1682}, {181, 612}, {200, 3169}, {511, 984}, {869, 2092}, {1253, 3270}, {1334, 2293}, {1742, 2808}, {2269, 2340}, {2323, 2330}, {2388, 2667}, {3057, 3059}

X(3688) = crosssum of X(7) and X(56)
X(3688) = crosspoint of X(8) and X(55)

### X(3689) = X(3218)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a - b - c)(2a - b - c)

X(3689) lies on these lines: {8, 2320}, {9, 55}, {37, 2177}, {42, 1100}, {44, 678}, {78, 3057}, {100, 518}, {214, 519}, {354, 1376}, {528, 908}, {612, 3290}, {650, 663}, {899, 1279}, {936, 3303}, {1104, 3214}, {1261, 2194}, {1318, 1320}, {1386, 3240}, {1837, 3189}, {2098, 2136}, {3058, 3452}

### X(3690) = X(1746)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b + c)2(- a2 + b2 + c2)

X(3690) lies on these lines: {3, 1796}, {9, 51}, {10, 3136}, {25, 220}, {31, 2273}, {37, 209}, {42, 213}, {55, 584}, {63, 295}, {71, 228}, {72, 306}, {101, 199}, {181, 756}, {184, 219}, {200, 1018}, {201, 1425}, {210, 430}, {212, 3270}, {373, 3305}, {511, 3219}, {612, 2295}, {1011, 3190}

X(3690) = crosspoint of X(71) and X(72)

### X(3691) = X(3294)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(ab + ac + 2bc)

X(3691) lies on these lines: {2, 1475}, {8, 9}, {10, 672}, {39, 899}, {41, 958}, {44, 2295}, {210, 1212}, {213, 1573}, {220, 2654}, {239, 1655}, {388, 966}, {405, 2280}, {519, 3294}, {604, 965}, {960, 2170}, {1018, 3626}, {1055, 2975}, {1107, 1193}, {1213, 2260}, {2276, 3214}, {3219, 3496}, {3501, 3617}

### X(3692) = X(1723)com(EXTOUCH TRIANGLE)

Trilinears    cot A cot2(A/2) : :      (Peter Moses, 12/19/11)
Barycentrics   a(b + c - a)2(b2 + c 2 - a2) : :
Barycentrics   (1 + cos A)(1 - sec A) : :

X(3692) lies on these lines: {1, 1257}, {6, 3693}, {8, 9}, {19, 5174}, {37, 4513}, {40, 5279}, {48, 4855}, {57, 4869}, {63, 69}, {78, 219}, {100, 610}, {145, 2257}, {190, 322}, {200, 1253}, {220, 3965}, {268, 271}, {281, 6735}, {282, 2057}, {312, 3305}, {326, 1332}, {341, 2322}, {380, 3871}, {519, 1723}, {579, 4684}, {595, 1743}, {644, 2324}, {1018, 1766}, {1034, 7080}, {1073, 3998}, {1229, 4384}, {1441, 3729}, {1445, 3912}, {1792, 1802}, {1809, 2289}, {2264, 3913}, {2285, 3501}, {2897, 3882}, {3436, 8804}, {3713, 4515}, {3731, 9623}, {3926, 7177}, {3949, 3984}, {3951, 4047}, {4073, 4319}, {10325, 10860}

X(3692) = isogonal conjugate of X(1435)
X(3692) = isotomic conjugate of X(1847)
X(3692) = crosspoint of X(i) and X(j) for these (i,j): {345, 1265}, {765, 1332}, {1016, 7256}
X(3692) = crosssum of X(i) and X(j) for these (i,j): {244, 6591}, {608, 1398}, {1015, 7250}
X(3692) = X(i)-cross conjugate of X(j) for these (i,j): {1260, 78}, {1802, 200}, {3694, 3692}
X(3692) = X(i)-Ceva conjugate of X(j) for these (i,j): {341, 200}, {345, 78}, {765, 4578}, {1792, 1260}
X(3692) = X(i)-beth conjugate of X(j) for these (i,j): {345, 6350}, {644, 1766}, {1332, 7013}, {3692, 219}, {6558, 3692}, {7259, 2324}
X(3692) = X(4462)-gimel conjugate of X(3692)
X(3692) = X(1)-zayin conjugate of X(1435)
X(3692) = isoconjugate of X(j) and X(j) for these (i,j): {1, 1435}, {2, 1398}, {4, 1407}, {6, 1119}, {7, 608}, {19, 269}, {25, 279}, {27, 1042}, {28, 1427}, {31, 1847}, {33, 738}, {34, 57}, {56, 278}, {65, 1396}, {81, 1426}, {85, 1395}, {92, 1106}, {108, 3669}, {158, 7099}, {162, 7216}, {196, 1413}, {208, 1422}, {222, 1118}, {225, 1412}, {244, 7128}, {270, 7147}, {273, 604}, {281, 7023}, {318, 7366}, {331, 1397}, {348, 7337}, {393, 7053}, {479, 607}, {648, 7250}, {667, 13149}, {934, 6591}, {1014, 1880}, {1088, 1973}, {1096, 7177}, {1262, 2969}, {1358, 7115}, {1416, 5236}, {1439, 5317}, {1440, 3209}, {1446, 2203}, {1461, 7649}, {1462, 1876}, {1474, 3668}, {2189, 6046}, {2207, 7056}, {2221, 7103}, {2489, 4616}, {3064, 6614}, {3213, 8809}, {6612, 7952}, {7339, 8735}, {7341, 8736}
X(3692) = barycentric product X(i)*X(j) for these {i,j}: {1, 1265}, {3, 341}, {8, 78}, {9, 345}, {10, 1792}, {21, 3710}, {33, 1264}, {55, 3718}, {63, 346}, {69, 200}, {72, 1043}, {75, 1260}, {76, 1802}, {77, 5423}, {210, 332}, {212, 3596}, {219, 312}, {220, 304}, {271, 7080}, {281, 3719}, {283, 3701}, {305, 1253}, {306, 2287}, {314, 2318}, {318, 1259}, {321, 2327}, {326, 7046}, {333, 3694}, {348, 728}, {394, 7101}, {480, 7182}, {521, 3699}, {522, 4571}, {525, 7259}, {644, 6332}, {645, 8611}, {646, 652}, {647, 7258}, {656, 7256}, {765, 2968}, {905, 6558}, {1098, 3695}, {1331, 4397}, {1332, 3239}, {1444, 4082}, {1809, 6735}, {1812, 2321}, {2289, 7017}, {2322, 3998}, {3270, 7035}, {3900, 4561}, {3926, 7079}, {3949, 7058}, {4025, 4578}, {4076, 7004}, {4163, 6516}, {4171, 4563}, {4391, 4587}, {4855, 6556}
X(3692) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1119}, {2, 1847}, {3, 269}, {6, 1435}, {8, 273}, {9, 278}, {31, 1398}, {33, 1118}, {41, 608}, {42, 1426}, {48, 1407}, {55, 34}, {63, 279}, {69, 1088}, {71, 1427}, {72, 3668}, {77, 479}, {78, 7}, {184, 1106}, {190, 13149}, {200, 4}, {201, 6046}, {210, 225}, {212, 56}, {219, 57}, {220, 19}, {222, 738}, {228, 1042}, {255, 7053}, {268, 1422}, {271, 1440}, {283, 1014}, {284, 1396}, {306, 1446}, {312, 331}, {326, 7056}, {341, 264}, {345, 85}, {346, 92}, {394, 7177}, {480, 33}, {521, 3676}, {577, 7099}, {603, 7023}, {612, 7103}, {644, 653}, {647, 7216}, {652, 3669}, {657, 6591}, {728, 281}, {810, 7250}, {906, 1461}, {1038, 7197}, {1040, 7195}, {1043, 286}, {1252, 7128}, {1253, 25}, {1259, 77}, {1260, 1}, {1264, 7182}, {1265, 75}, {1331, 934}, {1332, 658}, {1334, 1880}, {1792, 86}, {1802, 6}, {1812, 1434}, {1813, 4617}, {2175, 1395}, {2188, 1413}, {2193, 1412}, {2197, 7147}, {2212, 7337}, {2287, 27}, {2289, 222}, {2310, 2969}, {2318, 65}, {2324, 196}, {2327, 81}, {2328, 28}, {2332, 5317}, {2340, 1876}, {2638, 3937}, {2968, 1111}, {3119, 8735}, {3270, 244}, {3682, 1439}, {3689, 1877}, {3690, 1254}, {3693, 5236}, {3694, 226}, {3710, 1441}, {3713, 5307}, {3718, 6063}, {3719, 348}, {3781, 7204}, {3900, 7649}, {3939, 108}, {3949, 6354}, {3965, 1848}, {4055, 1410}, {4130, 3064}, {4171, 2501}, {4183, 8747}, {4319, 1851}, {4420, 7282}, {4433, 1874}, {4515, 1826}, {4558, 4637}, {4561, 4569}, {4563, 4635}, {4571, 664}, {4574, 1020}, {4578, 1897}, {4587, 651}, {4592, 4616}, {5227, 7365}, {5423, 318}, {6056, 603}, {6061, 270}, {6065, 7012}, {6516, 4626}, {6558, 6335}, {6602, 607}, {7004, 1358}, {7046, 158}, {7071, 1096}, {7074, 208}, {7079, 393}, {7080, 342}, {7085, 4320}, {7101, 2052}, {7256, 811}, {7258, 6331}, {7259, 648}, {7367, 7129}, {7368, 2331}, {8611, 7178}, {11517, 4341}

### X(3693) = X(672)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(b2 + c 2 - ab - ac)

X(3693) lies on these lines: {1, 728}, {2, 37}, {8, 1212}, {9, 55}, {44, 765}, {65, 3501}, {78, 220}, {100, 910}, {101, 2751}, {517, 1018}, {518, 672}, {522, 650}, {960, 1334}, {1026, 1642}, {1040, 2324}, {1155, 3509}, {1759, 3579}, {2329, 2646}, {3057, 3061}

### X(3694) = X(579)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(b + c - a)(b2 + c 2 - a2)

X(3694) lies on these lines: {1, 2336}, {8, 2335}, {9, 55}, {10, 37}, {71, 72}, {78, 219}, {281, 318}, {306, 307}, {518, 579}, {519, 1108}, {728, 2324}, {997, 2256}, {1761, 3579}, {1792, 2193}, {1793, 2327}, {2260, 3555}, {2345, 3085}, {3061, 3169}

### X(3695) = X(580)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)2(b2 + c 2 - a2)

X(3695) lies on these lines: {3, 345}, {4, 346}, {5, 312}, {8, 405}, {10, 37}, {12, 1089}, {40, 728}, {72, 306}, {78, 1062}, {100, 2915}, {190, 1330}, {304, 337}, {321, 442}, {519, 1104}, {1215, 3178}, {2049, 2345}, {3159, 3454}

### X(3696) = X(991)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(a2 - ab - ac - 2bc)
X(3696) = X(573)com[Inverse(n(Hexyl))]

X(3696) lies on these lines: {7, 8}, {10, 37}, {40, 1765}, {80, 2805}, {141, 1738}, {192, 3617}, {210, 321}, {239, 1386}, {318, 1882}, {536, 984}, {726, 3626}, {756, 3175}, {872, 3214}, {1150, 1155}, {1214, 2968}, {1376, 1402}, {1861, 1880}

X(3696) = midpoint of X(8) and X(75)

### X(3697) = X(631)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(b + c - a)(a^4 - 2a^2b^2 + b^4 + 6a^2bc - 4b^3c - 2a^2c^2 + 8b^2c^2 - 4bc^3 + c^4)   (Martin Sperr, March 23, 2015)
X(3697) = X(3522)com[Inverse(n(Hexyl))]

X(3697) lies on these lines: {2, 3555}, {8, 392}, {10, 12}, {37, 762}, {200, 405}, {474, 3361}, {517, 3091}, {518, 1698}, {756, 3214}, {936, 956}, {958, 3612}, {997, 1388}, {2551, 3419}, {3057, 3626}, {3219, 3579}, {3295, 3305}

### X(3698) = X(3522)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(a2 - b2 - c2 + 6bc)
X(3698) = X(631)com[Inverse(n(Hexyl))]

X(3698) lies on these lines: {2, 3057}, {8, 354}, {10, 12}, {55, 1706}, {281, 1888}, {374, 1828}, {474, 1319}, {517, 1656}, {518, 3617}, {936, 2099}, {958, 1155}, {1376, 2646}, {1836, 2551}, {1837, 2550}, {3555, 3626}

### X(3699) = X(1054)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/(b - c)

X(3699) lies on these lines: {1, 1120}, {2, 1280}, {8, 11}, {78, 341}, {100, 190}, {200, 312}, {210, 333}, {537, 1054}, {643, 645}, {644, 1639}, {658, 883}, {664, 668}, {765, 1331}, {1897,4033}, {2899, 3189}

X(3699) = isotomic conjugate of X(3676)
X(3699) = trilinear pole of line X(8)X(9)

### X(3700) = X(661)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b2 - c2)

X(3700) lies on these lines: {9, 1021}, {37, 647}, {101, 2689}, {321, 850}, {424, 2501}, {522, 650}, {523, 661}, {525, 1577}, {649, 900}, {693, 918}, {824, 3004}, {1635, 2490}, {2533, 3566}, {3910,4391}

X(3700) = isogonal conjugate of X(4565)
X(3700) = isotomic conjugate of X(4573)
X(3700) = perspector of hyperbola {{A,B,C,X(8),X(10)}}
X(3700) = intersection of trilinear polars of X(8) and X(10)

### X(3701) = X(3216)com(EXTOUCH TRIANGLE)

Barycentrics   bc(b + c)(b + c - a) : :

X(3701) lies on these lines: {1, 996}, {5, 3006}, {8, 210}, {10, 321}, {12, 313}, {76, 3263}, {281, 318}, {306, 857}, {442, 1230}, {612, 964}, {740, 3214}, {1826, 3610}, {2901, 3293}

X(3701) = isotomic conjugate of X(1014)

### X(3702) = X(3293)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c - a)(b + c - 2a)

X(3702) lies on these lines: {1, 321}, {8, 210}, {21, 261}, {75, 3616}, {78, 1229}, {306, 946}, {430, 1230}, {519, 1089}, {740, 1193}, {1043, 3615}, {1125, 1962}, {1441, 3485}

### X(3703) = X(31)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b2 + c2)

X(3703) lies on these lines: {2, 1390}, {5, 1089}, {8, 21}, {11, 312}, {38, 141}, {63, 3416}, {200, 1040}, {209, 306}, {321, 2886}, {346, 497}, {594, 2276}, {726, 2887}, {984, 1211}

### X(3704) = X(58)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b + c - a)(b2 + c2 + ab + ac)

X(3704) lies on these lines: {8, 21}, {10, 37}, {12, 321}, {40, 1503}, {65, 306}, {100, 1791}, {141, 986}, {197, 2915}, {312, 1329}, {346, 2551}, {524, 1046}, {960, 1682}, {1211, 2292}

### X(3705) = X(3550)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(2bc + ca + ab)

X(3705) lies on these lines: {1, 2}, {11, 312}, {63, 147}, {69, 1447}, {75, 325}, {183, 319}, {262, 321}, {318, 427}, {333, 2194}, {341, 1329}, {345, 497}, {982, 2887}, {1368, 2968}

### X(3706) = X(42)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b2 + c2 - bc)

X(3706) lies on these lines: {1, 2049}, {8, 210}, {38, 536}, {69, 1836}, {75, 354}, {306, 2886}, {321, 518}, {519, 1215}, {984, 3175}, {1043, 2646}, {1386, 3187}, {1456, 1943}, {3416, 3434}

### X(3707) = X(45)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(4a + b + c)

X(3707) lies on these lines: {6, 1125}, {8, 9}, {10, 44}, {37, 3244}, {45, 519}, {72, 374}, {142, 320}, {210, 3271}, {226, 1405}, {333, 645}, {966, 1698}, {1449, 3622}, {3247, 3623}

### X(3708) = X(150)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 - c2)2(b2 + c2 - a2)
X(3708) = X(101)com(Incentral)

X(3708) lies on these lines: {1, 163}, {19, 2159}, {31, 2157}, {115, 1365}, {512, 3022}, {523, 1146}, {525, 1565}, {774, 2179}, {811, 1821}, {1725, 1755}, {2170, 2611}, {2631, 2632}, {2642, 2643}

### X(3709) = X(798)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b + c - a)(b2 - c2)

X(3709) lies on these lines: {6, 2605}, {9, 3287}, {37, 523}, {44, 2609}, {101, 2701}, {213, 3049}, {512, 798}, {513, 665}, {522, 650}, {647, 661}, {657, 663}, {667, 2484}, {1919, 1960}

X(3709) = isogonal conjugate of X(4573)
X(3709) = complement of X(4374)

### X(3710) = X(1724)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b + c - a)(b2 + c2 - a2)

X(3710) lies on these lines: {8, 9}, {10, 321}, {71, 3610}, {72, 306}, {78, 345}, {145, 1453}, {201, 307}, {318, 341}, {519, 1724}, {946, 3006}, {1792, 1793}, {1834, 3175}

### X(3711) = X(3306)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(2b + 2c - a)

X(3711) lies on these lines: {8, 11}, {9, 55}, {45, 2177}, {214, 956}, {220, 3119}, {518, 3306}, {612, 1100}, {762, 2271}, {899, 3242}, {936, 3304}, {975, 2334}, {1376, 3218}

### X(3712) = X(896)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b2 + c2 - 2a2)

X(3712) lies on these lines: {8, 21}, {23, 100}, {37, 3291}, {101, 2768}, {109, 2760}, {522, 650}, {524, 896}, {528, 3006}, {536, 3011}, {846, 1211}, {2229, 2276}

### X(3713) = X(2285)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)2(a2 + ab + ac + 2bc)

X(3713) lies on these lines: {6, 8}, {9, 55}, {37, 78}, {72, 1766}, {219, 2321}, {220, 346}, {518, 2285}, {572, 956}, {608, 1861}, {958, 2268}, {1376, 1400}

### X(3714) = X(386)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b + c - a)(a2 + ab + ac + 2bc)

X(3714) lies on these lines: {4, 3416}, {8, 210}, {10, 37}, {12, 306}, {65, 321}, {72, 1089}, {318, 1859}, {536, 986}, {1220, 1999}, {1840, 1869}, {2292, 3175}

### X(3715) = X(3305)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(2b + 2c + a)

X(3715) lies on these lines: {6, 756}, {8, 3058}, {9, 55}, {10, 1836}, {42, 45}, {44, 612}, {518, 3305}, {748, 3242}, {940, 1757}, {960, 2098}, {1376, 3219}

### X(3716) = X(659)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)(b + c - a)(a2 - bc)

X(3716) lies on these lines: {2, 2254}, {11, 124}, {118, 120}, {522, 650}, {659, 812}, {676, 918}, {900, 3035}, {926, 3041}, {946, 2814}, {1960, 2787}

X(3716) = complement of X(2254)

### X(3717) = X(238)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b2 + c2 - ab - ac)

X(3717) lies on these lines: {1, 344}, {8, 9}, {10, 75}, {100, 2751}, {190, 516}, {200, 345}, {238, 519}, {908, 3006}, {1026, 1818}, {1220, 2983}

### X(3718) = X(1716)com(EXTOUCH TRIANGLE)

Trilinears        cot A csc A csc2(A/2) : cot B csc B csc2(B/2) : cot C csc C csc2(C/2)      (Peter Moses, 12/19/11)
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c - a)(b2 + c2 - a2)

X(3718) lies on these lines: {7, 3263}, {9, 312}, {10, 75}, {69, 72}, {78, 332}, {86, 975}, {190, 1760}, {305, 307}, {322, 668}, {740, 1716}

X(3718) = isogonal conjugate of X(1395)
X(3718) = isotomic conjugate of X(34)

### X(3719) = X(1711)com(EXTOUCH TRIANGLE)

Trilinears        cot(A/2) cot2A : cot(B/2) cot2B : cot(C/2) cot2C      (Peter Moses, 12/19/11)
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(b2 + c2 - a2)2

X(3719) lies on these lines: {9, 312}, {63, 69}, {78, 212}, {100, 2365}, {271, 1265}, {326, 394}, {346, 3219}, {740, 1711}, {2185, 3601}, {2269, 2339}

### X(3720) = X(756)com(INCENTRAL TRIANGLE)

Barycentrics    a(ab + ac + 2bc) : :

Let A'B'C' be the incentral triangle. Let A" be the intersection, other than the midpoint of BC, of the A-median and the bicevian ellipse of X(1) and X(2). Define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(3720). (Randy Hutson, December 26, 2015)

X(3720) lies on these lines: {1, 2}, {6, 748}, {11, 3136}, {31, 940}, {37, 38}, {55, 750}, {56, 1011}, {57, 968}, {73, 1985}, {77, 2898}, {81, 238}, {86, 310}, {171, 902}, {226, 1458}, {244, 1962}, {291, 1255}, {405, 1468}, {427, 2356}, {497, 2293}, {518, 756}, {649, 2666}, {799, 2107}, {846, 3218}, {851, 2646}, {942, 2292}, {960, 2650}, {991, 1699}, {1042, 3485}, {1100, 2238}, {1197, 3231}, {1279, 2239}, {1376, 2177}, {1818, 2886}, {1836, 3000}

X(3720) = {X(1),X(2)}-harmonic conjugate of X(42)
X(3720) = complement of X(4651)
X(3720) = bicentric sum of PU(84)
X(3720) = PU(84)-harmonic conjugate of X(649)

### X(3721) = X(213)com(INCENTRAL TRIANGLE)

Trilinears    b^3 + c^3 : :
Barycentrics    a(b + c)(b2 + c2 - bc)

X(3721) lies on these lines: {1, 32}, {6, 977}, {10, 762}, {37, 65}, {38, 1107}, {42, 2240}, {72, 2238}, {76, 1928}, {213, 758}, {257, 335}, {321, 1237}, {350, 695}, {712, 1930}, {737, 789}, {960, 3290}, {982, 2275}, {986, 2276}, {1574, 1739}, {1964, 2237}

X(3721) = crosssum of X(1) and X(32)

### X(3722) = X(3120)com(INCENTRAL TRIANGLE)

Barycentrics    a(2a2 + b2 + c2 - 2ab - 2ac) : :

X(3722) lies on these lines: {1, 88}, {31, 1331}, {37, 2246}, {38, 55}, {42, 1386}, {105, 612}, {200, 748}, {518, 896}, {528, 3120}, {614, 3158}, {756, 1621}, {899, 1279}, {968, 1282}, {976, 1036}, {1283, 2292}, {1496, 1795}, {1647, 3035}, {2346, 2648}

### X(3723) = X(1213)com(INCENTRAL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2a + 3b + 3c)
X(3723) = X(86)com[n(Medial)*n(Incentral)]

X(3723) lies on these lines: {1, 6}, {86, 536}, {519, 1213}, {551, 2321}, {583, 1334}, {594, 1125}, {902, 1962}, {910, 1953}, {941, 1392}, {966, 3241}, {1055, 2294}, {1319, 2171}, {1388, 2285}, {2178, 3295}, {2280, 3204}, {2345, 3622}, {2667, 3009}

### X(3724) = X(908)com(INCENTRAL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(b + c)(b2 + c2 - a2 + bc)
X(3724) = X(3218)com[n(Media)*n(Incentral)]

X(3724) lies on these lines: {1, 994}, {3, 2292}, {23, 1283}, {31, 48}, {36, 214}, {42, 181}, {55, 199}, {56, 2650}, {100, 740}, {187, 237}, {215, 2361}, {851, 1284}, {896, 3286}, {2078, 2611}, {2177, 2667}, {2198, 2200}

### X(3725) = X(312)com(INCENTRAL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(b + c)(b2 + c2 + ab + ac)
X(3725) = X(3210)com[n(Media)*n(Incentral)]

X(3725) lies on these lines: {1, 333}, {31, 48}, {37, 42}, {43, 312}, {55, 869}, {63, 2274}, {213, 1402}, {228, 1918}, {354, 1201}, {758, 982}, {960, 1193}, {2294, 3290}, {2308, 3248}

### X(3726) = X(2238)com(INCENTRAL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2abc - b3 - c3)

X(3726) lies on these lines: {{1, 32}, {37, 38}, {244, 1575}, {335, 350}, {518, 2238}, {519, 3125}, {758, 3230}, {902, 1962}, {940, 3218}, {942, 2295}, {982, 2276}

### X(3727) = X(2295)com(INCENTRAL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2abc + b3 + c3)

X(3727) lies on these lines: {1, 32}, {37, 1953}, {55, 3148}, {257, 350}, {517, 2295}, {762, 3626}, {960, 2238}, {986, 2275}, {1100, 2650}, {1107, 2170}, {1125, 3125}, {2276, 3061}

### X(3728) = X(192)com(INCENTRAL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(ab2 + ac2 + bc2 + cb2)
X(3728) = X(1278)com[n(Media)*n(Incentral)]

X(3728) lies on these lines: {8, 192}, {9, 2209}, {31, 1778}, {37, 42}, {38, 75}, {518, 2650}, {668, 1221}, {1107, 2309}, {1213, 3122}, {1573, 2388}, {2269, 2310}

X(3728) = anticomplement wrt incentral triangle of X(2667)

### X(3729) = X(3056)com(EXCENTRAL TRIANGLE)

Barycentrics   a2 - ab - ac + 2bc : :

X(3729) lies on these lines: {1, 87}, {2, 2415}, {6, 536}, {7, 346}, {8, 144}, {9, 75}, {10, 2996}, {37, 980}, {46, 1089}, {57, 312}, {63, 321}, {69, 527}, {76, 1423}, {78, 990}, {85, 728}, {86, 3247}, {101, 1958}, {141, 545}, {142, 344}, {148, 1654}, {193, 519}, {200, 1721}, {226, 345}, {239, 1278}, {341, 1706}, {385, 3550}, {522, 1027}, {664, 1419}, {940, 3175}, {975, 3159}, {1229, 1445}, {1909, 3208}, {1944, 2324}, {1975, 2329}, {2191, 2414}, {2999, 3210}

X(3729) = crosssum of PU(48)
X(3729) = isogonal conjugate of X(9315)
X(3729) = isotomic conjugate of X(9311)
X(3729) = anticomplement of X(3663)
X(3729) = crosspoint of PU'(48), where PU'(48) are the isogonal conjugates of PU(48)

### X(3730) = X(1212)com(EXCENTRAL TRIANGLE)

Barycentrics   a2(b2 + c2 - ab - ac + bc) : :

X(3730) lies on these lines: {1, 672}, {2, 2140}, {3, 101}, {4, 9}, {6, 595}, {8, 1018}, {35, 41}, {37, 579}, {39, 995}, {45, 2245}, {55, 218}, {76, 190}, {165, 170}, {213, 386}, {219, 572}, {284, 2911}, {348, 1025}, {514, 3177}, {517, 1212}, {519, 3208}, {644, 2975}, {813, 1083}, {910, 3579}, {993, 2329}, {1011, 3190}, {1030, 3204}, {1400, 3339}, {1726, 3219}, {1743, 2269}, {2203, 2328}, {2260, 3247}, {2267, 2323}, {2275, 3230}

X(3730) = isogonal conjugate of X(14377)
X(3730) = trilinear pole wrt 1st circumperp triangle of antiorthic axis
X(3730) = X(39)-of-excentral-triangle
X(3730) = Cundy-Parry Phi transform of X(103)
X(3730) = Cundy-Parry Psi transform of X(516)

### X(3731) = X(391)com(EXCENTRAL TRIANGLE)

Barycentrics a(a - 3b - 3c) : :

X(3731) lies on these lines: {1, 6}, {2, 2415}, {3, 1696}, {10, 346}, {35, 198}, {71, 2093}, {101, 2268}, {106, 2297}, {165, 846}, {200, 756}, {281, 1785}, {344, 3619}, {391, 519}, {573, 1334}, {610, 3465}, {612, 902}, {966, 2321}, {1400, 3339}, {1698, 1738}, {1707, 1961}, {1742, 3062}, {2182, 3612}, {2285, 3361}, {2999, 3305}

### X(3732) = X(3022)com(EXCENTRAL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 + b2 + c2 - 2bc)/(b - c)

X(3732) lies on these lines: {2, 1565}, {3, 3177}, {85, 169}, {100, 1292}, {101, 514}, {144, 153}, {150, 1146}, {190, 646}, {218, 3212}, {544, 1121}, {651, 653}, {673, 1111}

### X(3733) = X(656)com(TANGENTIAL TRIANGLE)

Barycentrics   a2(b - c)/(b + c) : :

X(3733) lies on these lines: {6, 798}, {28, 2401}, {36, 238}, {82, 876}, {86, 3253}, {99, 889}, {110, 901}, {512, 1326}, {523, 1325}, {649, 834}, {659, 3004}, {660, 662}, {665, 2483}, {669, 2106}, {688, 875}, {741, 2382}, {759, 2718}, {1474, 2424}

X(3733) = isogonal conjugate of X(3952)
X(3733) = X(92)-isoconjugate of X(4574)
X(3733) = pole wrt circumcircle of line X(1)X(21)
X(3733) = crossdifference of every pair of points on line X(10)X(37)

### X(3734) = X(6)com(1st BROCARD TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4 + 2b2c2

X(3734) = eigencenter of 1st Brocard triangle (Peter Moses, January 24, 2012); see X(3972)

Let A'B'C' be the 1st Brocard triangle. Let A" be the trilinear pole, with respect to A'B'C', of line BC, and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(3734).

X(3734) lies on these lines: {2, 99}, {4, 626}, {6, 538}, {30, 141}, {32, 76}, {39, 1975}, {69, 754}, {83, 194}, {182, 2782}, {183, 187}, {316, 3314}, {350, 2242}, {381, 625}, {1078, 3552}, {1235, 1968}, {1352, 2794}, {1909, 2241}

X(3734) = X(6) of 1st Brocard triangle
X(3734) = orthocentroidal-to-1st-Brocard similarity image of X(6)
X(3734) = anti-Artzt isogonal conjugate of X(598)
X(3734) = X(182)-of-anti-Artzt-triangle

### X(3735) = X(75)com(1st BROCARD TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b3 + c3 + abc)

X(3735) lies on these lines: {1, 32}, {2, 3125}, {6, 758}, {37, 517}, {38, 2170}, {39, 986}, {75, 712}, {76, 257}, {392, 3290}, {762, 3617}, {766, 3056}, {980, 1959}, {982, 1015}, {984, 1573}, {2549, 2795}, {2809, 3242}

X(3735) = X(75)-of-1st-Brocard-triangle

### X(3736) = X(2092)com(HEXYL TRIANGLE)

Barycentrics    a2(b2 + c2 + bc)/(b + c) : :

X(3736) lies on these lines: {1, 75}, {3, 6}, {21, 238}, {35, 1918}, {42, 81}, {43, 333}, {99, 731}, {110, 753}, {869, 984}, {872, 1757}, {995, 1001}, {1014, 1458}, {1408, 2594}, {1412, 1460}, {1707, 2258}, {1818, 2303}

X(3736) = crossdifference of every pair of points on line X(523)X(798)

### X(3737) = X(656)com(HEXYL TRIANGLE)

Trilinears    (b - c)(b + c - a)/(b + c) : :
Trilinears    1/(1 - cos(B - A) - cos(C - A) + cos(B - C)) : :
Trilinears    tan(B/2 - C/2) : :
Trilinears    tan(B' - C') : :, where A'B'C' = excentral triangle
Barycentrics   a(b - c)(b + c - a)/(b + c) : :

X(3737) lies on these lines: {1, 523}, {9, 3287}, {36, 238}, {54, 2616}, {59, 110}, {86, 1027}, {112, 2728}, {242, 514}, {284, 1024}, {512, 2959}, {521, 650}, {522, 663}, {741, 2726}, {759, 953}, {832, 1491}, {1443, 1447}

X(3737) = isogonal conjugate of X(4551)
X(3737) = SS(A->A') of X(2616), where A'B'C' is the excentral triangle

### X(3738) = X(2804)com(HEXYL TRIANGLE)

Barycentrics a(b - c)(b2 + c2 - a2 - bc) : :
Barycentrics    cos(A - B) - cos(A - C) : :

X(3738) lies on these lines: {1, 1769}, {11, 124}, {30, 511}, {100, 109}, {102, 104}, {117, 119}, {151, 153}, {656, 1955}, {885, 3254}, {1027, 1814}, {1317, 1361}, {3035, 3042}, {3036, 3040}

X(3738) = isogonal conjugate of X(2222)
X(3738) = X(522)-of-inner-Garcia triangle
X(3738) = X(2804)com(Excentral)

More Combos

Continuing the discussion of points Xcom(T), suppose that T is an arbitrary triangle, and let nT denote the normalization of T. Let NT denote the set of these triangles, as matrices, and let * denote matrix multiplication. Then NT is closed under *. Also, NT is closed under matrix inversion. Consequently, (NT, *) is a group, comparable to the group of stochastic matrices.

Suppose that T1 and T2 are triangles. In many cases, the product T1*T2 is well-defined (e.g., TCCT, page 175). However, n(T1*T2) may not be n(T1)*n(T2) if T1 and T2 are not normalized. Therefore, it is important, when dealing with products, to include the "n" if it is intended.

As a class of examples of triangles defined by matrix products, suppose that T1 is the cevian triangle of a triangle center f : g : h (barycentrics) and that T2 is the cevian triangle of a triangle center u : v : w. The A-vertex of the triangle T3 = (nT1)*(nT2) is given by

u(gu + hu + gv +hw) : hv(u + w) : gw(u + v),

from which it can be shown that T3 is perspective to the triangle ABC, with perspector

P = ghu(v + w) : hfv(w + u) : fgw(u + v).

Also, T3 is perspective to T2, with perspector

Q = u(v + w)(gu + hu + gv + hw) : v(w + u)(hv + fv + hw + fu) : w(u + v)(fw + gw + fu + gv).

Thus, T3 can be constructed directly from the pairs {P, ABC} and {Q, T2}. Regarding the possibility that T3 is also perspective to T1, the concurrence determinant for this condition factors as F1F2F3, where

F1 = u + v + w,
F2 = fu(g + h) + gv(h + f) + hw(f + g),
F3 = (fghuvw)2[(gu)-2 - (hu)-2 + (hv)-2 - (fv)-2 + (fw)-2 - (gw)-2].

Consequently, the perspectivity holds if and only if one of the three factors is 0, and from this result, various geometric results can be obtained.

Example: The Incentral and Medial triangles result by taking f : g : h = a : b : c and u : v : w = 1 : 1: 1. The product n(Incentral)*n(Medial) has vertices

A' = b + c : c : b        B' = c : c + a : a        C' = b : a : a + b

and its inverse has A-vertex A'' = bc + ca + ab : -bc - ca + ab : -bc + ca - ab, from which B'' and C'' are obtained cyclically.

The product n(Medial)*n(Incentral) has A-vertex a(2a + b + c) : b(c + a) : c(a + b), and the inverse of this product has A-vertex (a + b + c)(b + c) : -c(a + c) : -b(a + b).

Having considered the group NT of normalized triangles, we turn next to 3-point combos based on product triangles and inverse triangles. Recall that such a combo, denoted by Xcom(T), is defined, as in the preamble to X(3663), by the matrix product (x    y    z)*(nT), where nT is the normalization of T.

### X(3739) = X(37)com[n(INCENTRAL)*n(MEDIAL)]

Barycentrics    ab + ac + 2bc : :
X(3739) = =3*X(2) + X(75)

X(3739) lies on these lines: {2, 37}, {7, 966}, {10, 141}, {11, 126}, {44, 894}, {85, 1418}, {86, 239}, {241, 1441}, {274, 1107}, {320, 1654}, {335, 1268}, {740, 1125}, {742, 3008}, {872, 899}, {984, 1698}, {1010, 1104}, {1086, 1213}, {1368, 2886}, {1958, 2278}, {2234, 2309}

X(3739) = centroid of ABCX(75)
X(3739) = Kosnita(X(75),X(2)) point
X(3739) = complement of X(37)
X(3739) = anticomplement of X(4698)

### X(3740) = X(210)com[n(INCENTRAL)*n(MEDIAL)]

Barycentrics   a(b2 + c2 - ab - ac + 4bc) : :

In the plane of a triangle ABC, let
AaAbAc = A-extouch triangle, and define BaBbBc and CaCbCc cyclically
Ab' = AbAc∩BbBc, and define Bc' and Ca' cyclically
Ac' = AbAc∩CbCc, and define Ba' and Cb' cyclically
Then X(3740) = centroid-of-{Ab', Ac', Bc', Ba', Ca', Cb'}. These 6 points lie on the radical circle of the excircles. (Randy Hutson, September 14, 2016)

Let A'B'C' = excentral triangle. Let
Ab" = B'C'∩BbBc, and define Bc'' and Ca'' cyclically
Ac'' = B'C'∩CbCc, and define Ba'' and Cb'' cyclically
Then X(3740) = centroid-of-{Ab",Ac", Bc", Ba", Ca", Cb"}. These 6 points lie on an ellipse. (Randy Hutson, September 14, 2016)

Continuing, let Ab* = B'C'∩BcBa, and define Bc* and Ca* cyclically
Ac* = B'C'∩CaCb, and define Ba* and Cb* cyclically
Then X(3740) = centroid of {Ab*,Ac*,Bc*,Ba*,Ca*,Cb*}. These 6 points lie on the radical circle of the excircles. (Randy Hutson, September 14, 2016)

X(3740) lies on these lines: {2, 210}, {5, 10}, {9, 165}, {37, 43}, {44, 171}, {55, 3305}, {72, 1698}, {120, 125}, {200, 1001}, {374, 966}, {375, 511}, {405, 2900}, {612, 1386}, {748, 3246}, {756, 899}, {936, 958}, {1051, 1100}, {1107, 2664}, {1155, 3219}, {2801, 3035}, {3057, 3617}, {3555, 3624}

### X(3741) = X(321)com[n(INCENTRAL)*n(MEDIAL)]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2) + bc(b + c)

X(3741) lies on these lines: {1, 2}, {11, 1211}, {31, 1150}, {38, 321}, {75, 982}, {116, 126}, {141, 674}, {226, 1469}, {238, 333}, {256, 314}, {312, 984}, {516, 1764}, {518, 1215}, {594, 1575}, {964, 1468}, {993, 1011}, {2238, 2300}, {2260, 2345}, {2276, 2321}, {3136, 3454}

### X(3742) = X(3175)com[n(INCENTRAL)*n(MEDIAL)]

Barycentrics    a(b2 + c2 - ab - ac - 4bc)

Let A'B'C' be the intouch triangle. Let
AaAbAc = A-extouch triangle, and define BaBbBc and CaCbCc cyclically
Ab' = B'C'∩BbBc, and define Bc' and Ca' cyclically
Ac' = B'C'∩CbCc, and define Ba' and Cb' cyclically
Then X(3742) = centroid of {Ab',Ac',Bc',Ba',Ca',Cb'}. (Randy Hutson, September 14, 2016)

X(3742) lies on these lines: {1, 474}, {2, 210}, {31, 3246}, {37, 982}, {55, 3306}, {57, 1001}, {65, 3616}, {72, 3624}, {86, 1431}, {142, 2886}, {171, 1279}, {244, 1962}, {375, 2810}, {405, 3338}, {517, 549}, {614, 940}, {758, 942}, {958, 3333}, {1155, 1621}, {1698, 3555}, {3057, 3622}

X(3742) = complement of X(210)
X(3742) = X(551)com[Inverse(n(Hexyl))]

### X(3743) = X(10)com[n(MEDIAL)*n(INCENTRAL)]

Barycentrics   a(b + c)(a2 + b2 + c2 + 2ab + 2ac + bc) : :

X(3743) lies on these lines: {1, 21}, {10, 37}, {40, 2294}, {55, 2915}, {484, 1255}, {756, 3293}, {984, 2667}, {1126, 1757}, {1215, 3159}, {1486, 3295}, {3178, 3454}

X(3743) = complement of X(4647)
X(3743) = complement wrt incentral triangle of X(1)
X(3743) = perspector of Gemini triangle 12 and cross-triangle of Gemini triangles 11 and 12

### X(3744) = X(31)com[n(MEDIAL)*n(INCENTRAL)]

Barycentrics    a(2a2 + b2 + c2 - ab - ac)
X(3744) = X(2887)com(Incentral)

X(3744) lies on these lines: {1, 3}, {2, 1279}, {8, 1104}, {31, 518}, {37, 82}, {38, 902}, {42, 1386}, {58, 3555}, {63, 3052}, {72, 595}, {78, 1191}, {81, 643}, {145, 345}, {210, 238}, {612, 1001}, {614, 1376}, {748, 3246}, {749, 1100}, {960, 976}, {1455, 3476}, {2298, 2346}, {2886, 3011}, {2999, 3158}

### X(3745) = X(81)com[n(MEDIAL)*n(INCENTRAL)]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2a2 + b2 + c2 + ab + ac + 2bc)

X(3745) lies on these lines: {1, 3}, {2, 1386}, {6, 210}, {31, 37}, {33, 608}, {42, 1100}, {44, 756}, {81, 518}, {197, 2262}, {200, 1449}, {226, 1456}, {238, 1961}, {902, 1962}, {968, 3052}, {1279, 2239}, {2194, 2303}

### X(3746) = X(191)com[n(MEDIAL)*n(INCENTRAL)]

Trilinears    3 + 2 cos A : 3 + 2 cos B : 3 + 2 cos C      (Peter Moses, 12/19/11)
Barycentrics    a2(a2 - b2 - c2 - 3bc)

Let A'B'C' be the incentral triangle. Let La be the reflection of line BC in line B'C', and define Lb and Lc cyclically. Let A" = Lb∩Lc, B" = Lc∩La, C" = La∩Lb. The lines A'A", B'B", C'C" concur in X(3746). (Randy Hutson, September 14, 2016)

X(3746) lies on these lines: {1, 3}, {5, 3058}, {10, 1621}, {11, 3628}, {12, 546}, {21, 519}, {42, 595}, {58, 902}, {62, 1250}, {79, 516}, {80, 943}, {100, 1125}, {140, 3582}, {145, 993}, {191, 518}, {238, 3293}, {386, 2177}, {388, 3529}, {390, 1479}, {404, 551}, {442, 528}, {495, 3585}, {496, 632}, {497, 498}, {499, 1058}, {575, 2330}, {576, 3056}, {612, 1995}, {1001, 1698}, {1124, 3594}, {1126, 2308}, {1253, 1497}, {1283, 2292}, {1335, 3592}, {1376, 3624}, {1478, 3146}, {1500, 1914}, {2066, 3301}, {2241, 2276}, {2975, 3244}

X(3746) = incentral isogonal conjugate of X(1)
X(3746) = trilinear pole, wrt incentral triangle, of line that is incentral isogonal conjugate of incentral inellipse

### X(3747) = X(239)com[n(MEDIAL)*n(INCENTRAL)]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b + c)(a2 - bc)

X(3747) lies on these lines: {1, 21}, {6, 2667}, {9, 2209}, {37, 1918}, {42, 213}, {55, 869}, {100, 2664}, {187, 237}, {190, 714}, {238, 239}, {756, 3294}, {922, 3285}, {1914, 2210}, {2195, 3512}, {2245, 3122}, {2300, 2309}

X(3747) = isogonal conjugate of X(18827)
X(3747) = crossdifference of every pair of points on line X(2)X(661)

### X(3748) = X(1621)com[n(MEDIAL)*n(INCENTRAL)]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2a2 + b2 + c2 - 3ab - 3ac - 2bc)

X(3748) lies on these lines: {1, 3}, {37, 2280}, {42, 1279}, {105, 1255}, {210, 1001}, {226, 3058}, {390, 1836}, {518, 1621}, {672, 1100}, {910, 1953}, {954, 1864}, {968, 3242}, {1108, 2266}, {1362, 2772}, {1365, 3021}, {1453, 2334}, {1962, 2611}

### X(3749) = X(1707)com[n(MEDIAL)*n(INCENTRAL)]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(3a2 + b2 + c2 - 2ab - 2ac)

X(3749) lies on these lines: {1, 3}, {9, 983}, {31, 1331}, {43, 3158}, {63, 902}, {100, 614}, {200, 238}, {214, 1811}, {345, 519}, {518, 1707}, {612, 1621}, {943, 989}, {1279, 1376}, {1449, 2276}, {3011, 3434}

### X(3750) = X(846)com[n(MEDIAL)*n(INCENTRAL)]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 - 2ab - 2ac - bc)

X(3750) lies on these lines: {1, 3}, {2, 2177}, {42, 238}, {43, 1001}, {81, 902}, {99, 2668}, {256, 2346}, {333, 519}, {518, 846}, {748, 3240}, {941, 983}, {968, 984}

### X(3751) = X(1721)com[INVERSE(n(HEXYL TRIANGLE))]

Trilinears    aS - rSω : bS - rSω : cS - rSω
Barycentrics    a(a2 - b2 - c2 + 2ab + 2ac) : :
X(3751) = X(1) - 2 X(6)

Let A'B'C' denote the inverse of the normalized hexyl triangle. Then A', expressed in homogeneous barycentrics (i.e., not normalized), is given by

2a(b + c) : a2 + b2 - c2 : a2 - b2 + c2

In general, if a triangle DEF is perspective to a given triangle GHI from a point P and also perspector to another triangle JKL from a point Q, then, clearly, lines joining P to G, H, I, and line joining Q to J,K,L concur in pairs to form DEF. Thus, if constructions are known for G,H,I,P,J,K,I,Q, then a construction for DEF easily follows. In particular, the triangle A'B'C' can be constructed in several ways, as it is perspective to the following triangles with perspectors:

ABC from X(4), orthic triangle from X(4), medial triangle from X(2), intouch triangle from X(65), extouch triangle from X(10), Euler triangle X(4)

X(3751) lies on these lines: {1, 6}, {7, 1738}, {8, 193}, {10, 69}, {31, 1331}, {33, 1957}, {40, 511}, {42, 63}, {43, 57}, {46, 3293}, {55, 1707}, {58, 1792}, {78, 1468}, {81, 612}, {141, 1698}, {165, 1350}, {171, 200}, {182, 3576}, {210, 940}, {240, 2331}, {314, 989}, {355, 3564}, {386, 988}, {517, 1351}, {519, 1992}, {524, 3416}, {614, 3315}, {651, 2263}, {872, 2274}, {896, 2177}, {899, 3306}, {942, 1722}, {952, 1353}, {968, 3219}, {971, 1721}, {978, 3333}, {982, 2999}, {990, 1814}, {1125, 3618}, {1420, 1428}, {1445, 1458}, {1697, 3056}, {2330, 3601}, {2647, 3340}, {2648, 3577}, {2854, 2948}, {3094, 3097}, {3158, 3550}, {3216, 3338}, {3218, 3240}, {3589, 3624}

X(3751) = reflection of X(1) in X(6)
X(3751) = {X(1),X(6)}-harmonic conjugate of X(16475)

### X(3752) = X(995)com[INVERSE(n(HEXYL TRIANGLE))]

Barycentrics    a(b2 + c2 + ab + ac - 2bc) : :

X(3752) lies on these lines: {1, 474}, {2, 37}, {3, 1104}, {6, 57}, {31, 1155}, {34, 1466}, {38, 210}, {39, 1212}, {40, 1191}, {42, 244}, {43, 518}, {44, 63}, {55, 614}, {56, 197}, {65, 1193}, {72, 3216}, {81, 88}, {142, 2092}, {165, 3052}, {171, 1054}, {200, 3242}, {216, 1108}, {226, 1086}, {241, 2275}, {386, 942}, {517, 995}, {595, 3579}, {673, 893}, {940, 1100}, {958, 988}, {960, 978}, {980, 1107}, {1122, 2347}, {1201, 3057}, {1203, 3336}, {1210, 1834}, {1333, 1817}, {1403, 3185}, {1421, 3256}, {1435, 2331}, {1455, 1470}, {1616, 1697}, {1638, 3310}, {1738, 2886}, {1764, 2300}, {1876, 3192}, {3293, 3555}

### X(3753) = X(2)com[INVERSE(n(HEXYL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(a2 - b2 - c2 + 4bc)
X(3753) = X(376)com[Extouch]

X(3753) lies on these lines: {1, 474}, {2, 392}, {5, 1519}, {7, 3421}, {8, 443}, {9, 2093}, {10, 12}, {21, 3579}, {37, 1018}, {40, 405}, {46, 958}, {57, 956}, {80, 3255}, {142, 1145}, {318, 1148}, {354, 519}, {355, 377}, {374, 2835}, {375, 2390}, {404, 1385}, {406, 1902}, {475, 1829}, {518, 599}, {551, 2802}, {856, 1214}, {860, 1824}, {936, 3340}, {960, 1698}, {993, 1155}, {997, 2099}, {999, 3306}, {1012, 3359}, {1125, 3057}, {1378, 2362}, {1439, 1441}, {1737, 2886}, {1845, 3040}, {1861, 1905}, {2550, 3419}, {2650, 3214}

### X(3754) = X(5)com[INVERSE(n(HEXYL TRIANGLE))]

Barycentrics    a(b + c)(a2 - b2 - c2 + 3bc) : :
X(3754) = X(550)com[Extouch]

X(3754) lies on these lines: {1, 88}, {5, 2800}, {8, 2891}, {10, 12}, {21, 484}, {40, 1006}, {46, 993}, {80, 2475}, {140, 517}, {354, 3244}, {355, 2801}, {474, 2099}, {518, 3626}, {519, 942}, {551, 3057}, {860, 1825}, {997, 3340}, {1193, 1739}, {2295, 3125}, {2650, 3293}, {2975, 3336}, {3555, 3625}

X(3754) = X(389) of Fuhrmann triangle

### X(3755) = X(6)com[INVERSE(n(HEXYL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(3a2 + b2 + c2 - 2bc)
X(3755) = X(1350)com[Extouch]

X(3755) lies on these lines: {1, 142}, {4, 2331}, {6, 516}, {10, 37}, {40, 387}, {42, 226}, {43, 3452}, {65, 1439}, {386, 946}, {497, 2999}, {515, 990}, {519, 599}, {528, 1386}, {908, 3240}, {1001, 3008}, {1886, 2266}, {2177, 3011}

### X(3756) = X(106)com[INVERSE(n(HEXYL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (3a - b - c)(b - c)2
X(3756) = X(1293)com[Extouch]

Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Nagel line. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, B' = Mc∩Ma, C' = Ma∩Mb. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A"B"C" be the reflection of A'B'C' in the Nagel line. The triangle A"B"C" is homothetic to ABC, with center of homothety X(3756); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)

X(3756) lies on these lines: {1, 1145}, {2, 1280}, {8, 1120}, {11, 244}, {88, 149}, {106, 952}, {513, 1357}, {528, 1054}, {1015, 1146}, {1616, 1788}

X(3756) = centroid of (degenerate) pedal triangle of X(106)
X(3756) = crossdifference of every pair of points on line X(101)X(1293)

### X(3757) = X(846)com[INVERSE(n(1st BROCARD TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3 - a2b - a2c - b2c - bc2 - abc

X(3757) lies on these lines: {1, 2}, {25, 92}, {31, 894}, {55, 75}, {69, 3475}, {183, 322}, {192, 968}, {238, 1215}, {274, 2223}, {312, 1001}, {314, 2346}, {321, 1621}, {333, 518}, {675, 831}, {726, 846}, {1104, 1220}, {1281, 1283}, {1402, 1441}, {1909, 1965}

Central Triangles and More Combos

Many triangle centers can be defined (as just above) by the form Xcom(nT), where T denotes a central triangle. In order to present such centers, it is helpful to introduce the notation T(f(a,b,c), g(b,c,a)) for central triangles. Following TCCT, pages 53-54, suppose that each of f(a,b,c) and g(a,b,c) is a center-function or the zero function, and that one of these three conditions holds:

the degree of homogeneity of g equals that of f;
f is the zero function and g is not the zero function;
g is the zero function and f is not the zero function.

There are two cases to be considered: If g(a,b,c)=g(a,c,b), then the central triangle T(f(a,b,c), g(b,c,a)) is defined by the following 3x3 matrix (whose rows give homogeneous coordinates for the A-, B-, C- vertices, respectively):

f(a,b,c)     g(b,c,a)     g(c,a,b)
g(a,b,c)    f(b,c,a)      g(c,a,b)
g(a,b,c)    g(b,c,a)     f(c,a,b)

If g(a,b,c) is not equal to g(a,c,b), then the central triangle T(f(a,b,c),g(b,c,a)) is defined by the following 3x3 matrix:

f(a,b,c)     g(b,c,a)     g(c,b,a)
g(a,c,b)    f(b,c,a)      g(c,a,b)
g(a,b,c)    g(b,a,c)     f(c,a,b)

If the homogenous coordinates are chosen to be barycentric, as they are for present purposes, then the new notation for central triangles is typified by these examples:

Medial triangle = T(1, 0)
Anticomplementary triangle = T(-1, 1)
Incentral triangle = T(a, 0)
Excentral triangle = T(-a, b)
Euler triangle = T(2(b4 + c4 - b2c2 - c2a2 - a2b2), (b2 + c2 - a2)(a2 + b2 - c2))
2nd Euler triangle = T(2a2(b4 + c4 - 2b2c2 - c2a2 - a2b2), (a2 - b2 + c2)(a4 + b4 + c4 - 2a2c2 - 2b2c2))
3rd Euler triangle = T((b - c)2, c2 - a2 - bc)
4th Euler triangle = T((b + c)2, c2 - a2 + bc)
5th Euler triangle = T(2(b2 + c2), b2 + c2 - a2 )

The vertices of the five Euler triangles lie on the nine-point circle. (See C. Kimberling, "Twenty-one points on the nine-point circle," Mathematical Gazette 92 (2008) 29-38.) Randy Hutson observes (January 8, 2015) that the 2nd Euler triangle is the complement of the circumorthic triangle, the 2nd Euler triangle is the reflection of the orthic triangle in X(5); the 3rd Euler triangle is the complement of the 1st circumperp triangle, the 4th Euler triangle is the complement of the 2nd circumperp triangle; the 4th Euler triangle the triangle whose sidelines are the radical axes of each of the Odehnal tritangent circles (defined at X(6176)) and the corresponding excircle; the 5th Euler triangle is the complement of the circummedial triangle, and the vertices of the 5th Euler triangle are the intersections, other than midpoints of the sides of ABC, of the nine-point circle and the medians.

The notation T(f(a,b,c),g(b,c,a)) can be used to define several more triangles; in each case, two of the perspectivities can be used to construct the triangle.

T(bc, b2) is perspective to ABC with perspector X(6); other such pairs: incentral, X(192), excentral, X(1045); tangential, X(6); anticomplementary, X(192)

T(-bc, b2) is perspective to ABC with perspector X(6); incentral, X(2); tangential, X(6); T(bc,b2), X(6)

T(bc, c2) is perspective to ABC with perspector X(76); anticomplementary, X(8); 4th Brocard, X(76), Fuhrmann, X(8); cevian(X(350)),X(1)

T(-bc, c2) is perspective to ABC with perspector X(76); cevian(X(321)),X(75); anticevian(X(8)), X(2)

T(a2, a2 - b2) is perspective to the medial triangle with perspector X(3); tangential , X(3); symmedial, X(2); cevian(X(287)), X(98)

T(a2, a2 - c2) is perspective to the tangential triangle with perspector X(25); orthic , X(25); medial, X(6); 1st Lemoine, X(1383); circummedial, X(251)

T(a2, a2 + b2) is perspective to ABC with perspector X(83); medial, X(6); 1st Neuberg, X(182)

T(a2, a2 + c2) is perspective to ABC with perspector X(141); medial, X(3); 1st Neuberg, X(182); T(a2,a2 - c2), X(2); cevian(X(69)), X(6); cevian(X(76)), X(2); cevian(X(3313)),X(39)

The triangles IT1 and IT2 are triply perspective to ABC (as are T1 and T3). Order-label IT1 as A'B'C'. The three perspectivities are then given by

AA'∩BB'∩CC' = X(3862)
AB'∩BC'∩CA' = f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ab(a2 - bc)(c2 - ab)(b2 + bc + c2)
AC'∩BA'∩CB' = f(a,c,b) : f(b,a,c) : f(c,b,a)

Peter Moses noted (12/23/2011) that T(bc, b2) is triply perspective to ABC. With T(bc, b2) order-labeled as A'B'C', the three perspectivities with perspectors are as follows:

AA'∩BB'∩CC' = X(6); AB'∩BC'∩CA' = P(6); AC'∩BA'∩CB' = U(6). (The notation P(k) and U(k) refers to a bicentric pair of points; see Tables at the top of ETC.)
Likewise, T(bc, c2) is triply perspective to ABC, with perspectors: X(76), P(10), and U(10).

(This section was added to ETC on 12/23/2011.)

### X(3758) = X(3661)com[T(bc, b2)]

Barycentrics   2a2 + bc : 2b2 + ca : 2c2 + ab

X(3758) lies on these lines: {1, 190}, {2, 44}, {6, 75}, {7, 3618}, {8, 1992}, {9, 86}, {41, 662}, {43, 2234}, {81, 312}, {83, 3673}, {85, 651}, {87, 1964}, {192, 1100}, {193, 319}, {256, 749}, {291, 751}, {318, 648}, {524, 3661}, {594, 3629}, {597, 1086}, {645, 1509}, {750, 3570}, {872, 1740}, {1449, 3644}, {1760, 2285}, {3589, 3662}

### X(3759) = X(3662)com[T(-bc, b2)]

Barycentrics   -2a2 + bc : -2b2 + ca : -2c2 + ab

X(3759) lies on these lines: 1, 872}, {2, 319}, {6, 75}, {7, 1992}, {8, 1386}, {43, 1964}, {44, 192}, {86, 1449}, {87, 2234}, {145, 344}, {190, 1743}, {193, 320}, {256, 751}, {273, 648}, {291, 749}, {312, 3187}, {524, 3662}, {594, 597}, {604, 662}, {664, 1445}, {1043, 1453}, {1086, 3629}, {1740, 3248}, {1760, 2082}, {3589, 3661}

### X(3760) = X(2275)com[T(-bc, c2)]

Barycentrics   bc(a2 - 2bc) : ca(b2 - 2ca) : ab(c2 - 2ab)

X(3760) lies on these lines: {1, 76}, {33, 1235}, {35, 183}, {36, 1975}, {69, 1479}, {75, 1089}, {172, 3734}, {274, 3624}, {304, 1111}, {312, 1930}, {313, 3293}, {315, 3583}, {339, 1062}, {384, 609}, {538, 2275}, {668, 3632}

X(3760) = isotomic conjugate of X(749)
X(3760) = {X(1),X(76)}-harmonic conjugate of X(3761)

### X(3761) = X(2276)com[T(bc, c2)]

Barycentrics   bc(a2 + 2bc) : ca(b2 + 2ca) : ab(c2 + 2ab)

X(3761) lies on these lines: {1, 76}, {34, 1235}, {35, 1975}, {36, 183}, {43, 310}, {69, 1478}, {75, 537}, {85, 1930}, {274, 1698}, {304, 1089}, {315, 3585}, {339, 1060}, {385, 609}, {538, 2276}, {693, 1022}, {975, 1228}, {1018, 3729}, {1914, 3734}

X(3761) = isotomic conjugate of X(751)
X(3761) = {X(1),X(76)}-harmonic conjugate of X(3760)

### X(3762) = X(1491)com[T(bc, c2)]

Barycentrics   bc(b - c)(b + c - 2a) : ca(c - a)(c + a - 2b) : ab(a - b)(a + b - 2c)

X(3762) lies on these lines: {1, 3716}, {4, 2814}, {10, 2254}, {80, 3738}, {153, 2827}, {190, 646}, {514, 661}, {659, 2787}, {900, 1145}, {918, 1086}, {1121, 2481}

X(3762) = isotomic conjugate of X(3257)
X(3762) = X(663)-of-inner-Garcia triangle

### X(3763) = X(3618)com[T(a2, a2 + c2)]

Barycentrics   a2 + 2b2 + 2c2 : b2 + 2c2 + 2a2 : c2 + 2a2 + 2b2

X(3763) lies on these lines: {2, 6}, {3, 2916}, {5, 1350}, {10, 3242}, {49, 182}, {66, 154}, {125, 2930}, {140, 1352}, {159, 1853}, {316, 2076}, {381, 3098}, {511, 1656}, {518, 1698}, {631, 1503}, {632, 3564}, {1086, 2345}, {1125, 3416}, {1386, 3624}, {2097, 3452}

### X(3764) = X(869)com[T(bc, b2)]

Barycentrics   a2(b3 + c3 + abc) : b2(c3 + a3 + abc) : c2(a3 + b3 + abc)

X(3764) lies on these lines: {1, 3122}, {2, 2228}, {6, 560}, {31, 2245}, {39, 3271}, {42, 2183}, {75, 256}, {291, 751}, {320, 982}, {511, 2274}, {573, 1918}, {674, 869}, {1837, 2310}, {1964, 2277}, {2275, 3248}

### X(3765) = X(869)com[T(bc, c2)]

Barycentrics   bc(a3 + b2c + c2b) : ca(b3 + c2a + a2c) : ab(c3 + a2b + b2a)

X(3765) lies on these lines: {2, 330}, {4, 8}, {6, 313}, {75, 1654}, {76, 239}, {668, 3661}, {730, 869}, {894, 3596}, {1230, 3187}

### X(3766) = X(3250)com[T(bc, c2)]

Barycentrics   bc(b - c)(bc - a2) : ca(c - a)(ca - b2) : ab(a - b)(ab - c2)

X(3766) lies on these lines: {2, 665}, {75, 900}, {150, 928}, {244, 1111}, {316, 512}, {334, 876}, {335, 918}, {513, 3261}, {514, 661}, {668, 891}, {885, 2481}

X(3766) = isotomic conjugate of X(660)

### X(3767) = X(3053)com[T(a2, a2 - c2)]

Barycentrics   a4 + (b2 - c2)2 : b4 + (c2 - a2)2 : c4 + (a2 - b2)2

X(3767) is the center of the inellipse that is the barycentric square of the orthic axis. The Brianchon point (perspector) of the inellipse is X(393). (Randy Hutson, October 15, 2018)

X(3767) lies on these lines: {2, 39}, {3, 230}, {4, 32}, {5, 6}, {20, 187}, {25, 2353}, {30, 3053}, {53, 1598}, {69, 626}, {148, 3552}, {172, 1478}, {193, 625}, {216, 3547}, {232, 3542}, {235, 2138}, {315, 385}, {382, 1384}, {388, 2242}, {393, 800}, {427, 1184}, {497, 2241}, {498, 2276}, {499, 2275}, {574, 631}, {609, 3585}, {637, 1504}, {638, 1505}, {946, 1572}, {1015, 3086}, {1204, 1562}, {1368, 1611}, {1479, 1914}, {1500, 3085}, {1506, 3090}, {1714, 2238}, {1834, 2271}, {1899, 2450}, {1976, 2909}, {2023, 3095}, {3054, 3526}

X(3767) = complement of X(3926)
X(3767) = anticomplement of X(3788)
X(3767) = {X(5),X(6)}-harmonic conjugate of X(2548)

### X(3768) = X(3250)com[T(bc, b2)]

Barycentrics   a2(b - c)(2bc - ab - ac) : b2(c - a)(2ca - bc - ba) : c2(a - b)(2ab - ca - cb)

X(3768) lies on these lines: {44, 513}, {190, 646}, {292, 3572}, {573, 2827}, {813, 3257}, {888, 2667}, {1015, 1960}

X(3768) = isogonal conjugate of X(4607)

### X(3769) = X(3705)com[T(-bc, b2)]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2a3 + abc - b2c - bc2

X(3769) lies on these lines: {2, 1386}, {31, 312}, {43, 1964}, {55, 1999}, {75, 171}, {100, 2352}, {145, 2646}, {190, 1707}, {239, 1376}, {333, 612}, {740, 3550}, {940, 3757}, {1155, 3210}, {1748, 1897}, {2162, 2235}, {3052, 3685}

### X(3770) = X(3736)com[T(bc, c2)]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(a3 + abc + b2c + bc2)

X(3770) lies on these lines: {6, 76}, {37, 1655}, {75, 1654}, {81, 1230}, {99, 1030}, {239, 1269}, {256, 714}, {274, 1213}, {310, 2238}, {313, 894}, {314, 524}, {319, 321}, {350, 1100}, {384, 2220}, {385, 1333}, {538, 2092}, {594, 668}, {1228, 2303}, {1330, 3416}

### X(3771) = X(1707)com[T(a2, a2 - b2)]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3 + b3 + c3 - a2b - a2c

X(3771) lies on these lines: {1, 2}, {55, 2887}, {345, 726}, {752, 3052}, {1575, 3694}

### X(3772) = X(1707)com[T(a2, a2 - c2)]

Barycentrics    a3 + b3 + c3 - b2c - c2b : :

X(3772) is the center of the inconic that is the polar conjugate of the isotomic conjugate of the incircle. (Randy Hutson, October 15, 2018)

X(3772) lies on these lines: {1, 442}, {2, 37}, {4, 1104}, {6, 226}, {11, 33}, {27, 1333}, {31, 1836}, {44, 329}, {55, 3011}, {56, 225}, {57, 1020}, {72, 1714}, {172, 379}, {220, 3008}, {223, 3554}, {278, 393}, {387, 3487}, {475, 3086}, {497, 1279}, {516, 3052}, {528, 3749}, {851, 2352}, {899, 2318}, {908, 2911}, {946, 1191}, {1072, 3428}, {1125, 2049}, {1201, 2654}, {1284, 3185}, {1329, 1722}, {1376, 1738}, {1612, 3651}, {2218, 3145}, {2887, 3416}, {2999, 3553}, {3191, 3216}, {3434, 3744}

X(3772) = trilinear pole of the polar, wrt the Fuhrmann circle, of the perspector of the Fuhrmann circle

### X(3773) = X(141)com[INVERSE(nT(bc, c2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b2 + c2 + bc)

X(3773) lies on these lines: {8, 238}, {10, 37}, {141, 726}, {306, 1215}, {313, 1089}, {319, 1757}, {321, 2887}, {519, 597}, {752, 3416}, {984, 3661}, {3703, 3741}

### X(3774) = X(39)com[INVERSE(nT(bc, b2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(b + c)(b2 + c2 + bc)

X(3774) lies on these lines: {10, 37}, {39, 518}, {42, 3121}, {100, 1908}, {192, 1921}, {213, 872}, {984, 2276}, {1045, 3508}, {1574, 3739}

### X(3775) = X(594)com[INVERSE(nT(bc, c2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (2a + b + c)(b2 + c2 + bc)

X(3775) lies on these lines: {1, 319}, {10, 141}, {11, 1211}, {238, 1654}, {594, 726}, {984, 3661}, {1100, 1125}, {2308, 3578}

X(3775) = reflection of X(1100) in X(1125)
X(3775) = complement of X(4649)

### X(3776) = X(3700)com[INVERSE(nT(-bc, c2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)(b2 + c2 - bc)

X(3776) lies on these lines: {241, 514}, {321, 693}

X(3776) = isotomic conjugate of X(4621)

### X(3777) = X(1577)com[INVERSE(nT(-bc, c2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b - c)(b2 + c2 - bc)

X(3777) lies on these lines: {10, 514}, {513, 663}, {659, 905}, {891, 1734}

### X(3778) = X(1964)com[INVERSE(nT(-bc, b2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b + c)(b2 + c2 - bc)

X(3778) lies on these lines: {6, 560}, {10, 3728}, {31, 579}, {37, 3122}, {39, 2309}, {42, 181}, {63, 1716}, {71, 3747}, {75, 700}, {141, 2228}, {142, 244}, {209, 3725}, {256, 291}, {313, 714}, {573, 2209}, {674, 1964}, {723, 789}, {749, 751}, {869, 2277}, {982, 2887}, {1445, 1758}, {1918, 2245}, {2085, 3670}, {2275, 3056}, {3009, 3688}, {3123, 3663}

### X(3779)  =  X(3056)com[T(-bc, b2)]

Barycentrics    a2(b3 + c3 - ab2 - ac2 - abc) : :

Let A'B'C' be the extangents triangle. Let A" be the trilinear pole, wrt A'B'C', of line BC, and define B" and C" cyclically. Let A* be the trilinear pole, wrt A'B'C', of line B"C", and define B* and C* cyclically. The lines A'A*, B'B*, C'C* concur in X(3779). (Randy Hutson, December 2, 2017)

X(3779) lies on these lines: {1, 3688}, {6, 31}, {7, 8}, {19, 1843}, {40, 511}, {56, 1818}, {181, 200}, {210, 966}, {269, 1362}, {291, 1740}, {579, 2223}, {869, 2277}, {968, 3690}, {1400, 2340}, {1402, 3190}, {1486, 2911}, {1631, 2174}, {1654, 3681}, {1721, 2808}, {1743, 3271}, {1964, 2275}, {2093, 2810}, {2294, 3728}, {3169, 3174}

X(3779) = homothetic center of extangents triangle and reflection of intangents triangle in X(6)
X(3779) = perspector of extangents triangle and Mandart-excircles triangle
X(3779) = X(7)-of-extangents-triangle if ABC is acute

### X(3780) = X(3721)com[T(-bc, b2)]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2b + a2c + 2abc - b2c - bc2)

X(3780) lies on these lines: {1, 1573}, {6, 8}, {37, 3691}, {39, 3293}, {42, 1107}, {43, 2275}, {44, 1334}, {72, 3727}, {145, 2176}, {172, 3684}, {213, 519}, {239, 1909}, {518, 3721}, {956, 2271}, {992, 1100}, {1015, 3216}, {1230, 3187}, {1475, 1575}, {1724, 2241}, {1743, 3208}, {3230, 3244}, {3555, 3726}

### X(3781) = X(9)com[INVERSE(nT(bc, c2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b2 + c2 - a2)(b2 + c2 + bc)

X(3781) lies on these lines: {1, 3688}, {3, 48}, {9, 511}, {35, 2175}, {51, 3305}, {58, 2273}, {63, 295}, {69, 72}, {182, 2323}, {200, 3033}, {209, 940}, {220, 1350}, {238, 3056}, {386, 2300}, {517, 2550}, {674, 1001}, {936, 970}, {984, 1469}, {991, 3730}, {2276, 3736}, {2979, 3219}, {3098, 3220}

### X(3782) = X(2887)com[T(-bc, c2)]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b3 + c3 + ab2 + ac2 - b2c - c2b)

X(3782) lies on these lines: {1, 30}, {2, 45}, {5, 3670}, {7, 940}, {11, 982}, {12, 986}, {38, 2886}, {75, 1211}, {141, 321}, {210, 1738}, {226, 1465}, {306, 536}, {312, 3662}, {320, 1999}, {516, 3744}, {524, 3187}, {726, 2887}, {908, 3752}, {1266, 3687}, {1699, 3677}, {3242, 3434}

X(3782) = pole of de Longchamps line wrt incircle

### X(3783) = X(668)com[T(bc, c2)]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 - bc)(b2 + c2 + bc)

X(3783) lies on these lines: {1, 2}, {69, 1740}, {72, 695}, {87, 193}, {100, 2239}, {238, 1914}, {291, 518}, {319, 1964}, {320, 2234}, {350, 740}, {668, 730}, {672, 1282}, {752, 2230}, {788, 1491}, {984, 2276}, {1654, 2309}

### X(3784) = X(57)com[INVERSE(nT(-bc, c2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b2 + c2 - a2)(b2 + c2 - bc)

X(3784) lies on these lines: {1, 1401}, {3, 73}, {36, 1397}, {51, 3306}, {57, 511}, {63, 295}, {165, 1362}, {171, 1469}, {182, 2003}, {200, 2810}, {394, 1473}, {517, 3474}, {982, 3056}, {1040, 1364}, {1350, 1407}, {2979, 3218}

### X(3785) = X(2548)com[INVERSE(nT(a2, a2 + b2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 + c2 - a2)(b2 + c2 + 3a2)

X(3785) lies on these lines: {2, 32}, {3, 69}, {4, 183}, {20, 76}, {39, 193}, {99, 3522}, {140, 1007}, {141, 3053}, {187, 3620}, {217, 394}, {316, 3091}, {317, 3088}, {325, 631}, {376, 1975}, {1384, 3619}

X(3785) = isotomic conjugate of X(8801)
X(3785) = anticomplement of X(2548)

### X(3786) = X(1654)com[INVERSE(nT(bc, c2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a + b)(a + c)(a - b - c)(b2 + c2 + bc)

X(3786) lies on these lines: {1, 3728}, {8, 314}, {9, 21}, {29, 1827}, {58, 1757}, {72, 894}, {81, 612}, {86, 518}, {210, 333}, {228, 3219}, {511, 1654}, {869, 984}, {960, 1043}, {1045, 2292}

### X(3787) = X(1196)com[INVERSE(nT(a2, a2 - c2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b2 + c2)(b2 + c2 - 3a2)

X(3787) lies on these lines: {32, 394}, {39, 3051}, {51, 3231}, {184, 187}, {323, 1627}, {511, 1196}, {800, 3289}, {1194, 2979}, {1351, 1611}, {1501, 3292}, {1692, 1993}, {2056, 2076}, {3053, 3167}, {3060, 3291}}

### X(3788) = X(3053)comT(a2, a2 - b2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a4 + b4 + c4 - a2b2 - a2c2)

X(3788) lies on these lines: {2, 39}, {3, 114}, {4, 625}, {5, 3734}, {32, 325}, {69, 1692}, {115, 1975}, {140, 141}, {187, 315}, {316, 3552}, {754, 3053}, {1007, 2548}, {1078, 3314}

X(3788) = complement of X(3767)
X(3788) = anticomplement of X(7886)

### X(3789) = X(3679)com[INVERSE(nT(bc, c2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 - ab - ac - 2bc)(b2 + c2 + bc)

X(3789) lies on these lines: {1, 1573}, {2, 210}, {8, 350}, {9, 1282}, {10, 2140}, {120, 141}, {984, 2276}, {1001, 2280}, {1211, 2886}, {1654, 3056}, {3452, 3741}

### X(3790) = X(3662)com[INVERSE(nT(bc, c2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b2 + c2 + bc)

X(3790) lies on these lines: {8, 9}, {10, 192}, {11, 312}, {190, 3416}, {341, 3704}, {495, 3695}, {726, 3662}, {984, 3661}, {1278, 1738}, {3596, 3701}, {3681, 3690}

### X(3791) = X(2887)com[T(-bc, b2)]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b2 + c2 + bc)

X(3791) lies on these lines: {1, 333}, {6, 1215}, {10, 3745}, {31, 740}, {43, 1964}, {171, 239}, {238, 1999}, {321, 2308}, {519, 3703}, {1386, 3741}, {3244, 3748}

### X(3792) = X(190)com[INVERSE(nT(bc, c2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a2 - b2 - c2 + bc)(b2 + c2 + bc)

X(3792) lies on these lines: {1, 674}, {31, 2979}, {36, 2245}, {238, 511}, {320, 758}, {748, 3060}, {788, 1491}, {984, 1469}, {1216, 3072}, {1944, 2607}, {2175, 3098}

### X(3793) = X(625)com[INVERSE(nT(a2, a2 - c2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 + c2 - 3a2)(b2 + c2)

X(3793) lies on these lines: {3, 193}, {30, 148}, {32, 141}, {69, 1384}, {187, 524}, {194, 548}, {230, 625}, {441, 3580}, {574, 3629}, {2080, 3564}

### X(3794) = X(2)com[INVERSE(nT(-bc, c2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a + b)(a + c)(b + c - a)(b2 + c2 - bc)

X(3794) lies on these lines: {1, 21}, {2, 51}, {29, 1828}, {86, 1431}, {210, 333}, {404, 1730}, {1010, 3753}, {1330, 2478}, {1403, 3286}, {3056, 3705}

### X(3795) = X(3097)com[INVERSE(nT(bc, b2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2a2 + ab + ac - bc)(b2 + c2 + bc)

X(3795) lies on these lines: {1, 1575}, {2, 740}, {6, 2108}, {43, 55}, {100, 985}, {291, 1002}, {518, 3097}, {678, 3240}, {984, 2276}

### X(3796) = X(2)com[INVERSE(nT(a2, a2 + c2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b2 + c2 - a2)(b2 + c2 + 3a2)

X(3796) lies on these lines: {2, 154}, {3, 49}, {6, 22}, {25, 182}, {206, 1619}, {458, 1629}, {1184, 1691}, {1350, 1993}, {3051, 3053}

X(3796) = isogonal conjugate of X(8801)
X(3796) = crossdifference of every pair of points on line X(826)X(2501)

### X(3797) = X(2)com[INVERSE(nT(bc, b2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 - bc)(b2 + c2 + bc)

X(3797) lies on these lines: {2, 37}, {190, 742}, {194, 304}, {238, 239}, {335, 726}, {518, 2113}, {824, 3250}, {984, 3661}

### X(3798) = X(3239)com[INVERSE(nT(a2, a2 - c2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)(b2 + c2 - 3a2)

X(3798) lies on these lines: {239, 514}, {812, 3676}, {900, 2487}, {2254, 3667}, {2488, 3309}, {2786, 3239}

### X(3799) = X(2)com[INVERSE(nT(bc, c2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a - b)(a - c)(b2 + c2 + bc)

X(3799) lies on these lines: {100, 101}, {190, 513}, {668, 891}, {2802, 3679}, {2809, 3681}, {3507, 3747}

### X(3800) = X(512)com[INVERSE(nT(a2, a2 + b2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 - c2)(b2 + c2 + 3a2)

X(3800) lies on these lines: {30, 511}, {419, 2501}, {879, 3527}, {1734, 3004}

X(3800) = isogonal conjugate of X(907)

### X(3801) = X(2533)com[INVERSE(nT(-bc, b2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 - c2)(b2 + c2 - bc)

X(3801) lies on these lines: {514, 659}, {523, 656}, {826, 1089}

### X(3802) = X(1909)com[INVERSE(nT(bc, c2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 - bc)2(b2 + c2 + bc)

X(3802) lies on these lines: {1, 39}, {238, 239}, {1682, 3022}

### X(3803) = X(905)com[INVERSE(nT(a2, a2 + c2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b - c)(3a2 + b2 + c2)

X(3803) lies on these lines: {36, 238}, {649, 3309}, {650, 830}

### X(3804) = X(647)com[INVERSE(nT(a2, a2 + c2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b2 - c2)(3a2 + b2 + c2)

X(3804) lies on these lines: {187, 237}, {1499, 3265}, {2525, 3566}

### X(3805) = X(2786)com[INVERSE(nT(bc,c2))]

Barycentrics    a(b - c)(a2 + bc)(b2 + c2 + bc) : :

X(3805) is the infinite point of perspectrices of every pair of {ABC, Gemini triangle 32, Gemini triangle 34}. (Randy Hutson, January 15, 2019)

X(3805) lies on these lines: {30, 511}, {38, 661}

X(3805) = isogonal conjugate of X(30670)
X(3805) = crossdifference of every pair of points on line X(6)X(256)
X(3805) = ideal point of PU(35)

### X(3806) = X(2525)com[INVERSE(nT(a2, a2 + c2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b4 - c4)(3a2 + b2 + c2)

X(3806) lies on these lines: {230, 231}, {826, 2474}

### X(3807) = X(2525)com[INVERSE(nT(bc, b2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b)(a - c)(b2 + c2 + bc)

X(3807) lies on these lines: {100, 190}, {514, 668}

### X(3808) = X(2786)com[INVERSE(nT(-bc, c2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b - c)(a2 - bc)(b2 + c2 - bc)

X(3808) lies on this line: {30, 511}

X(3808) = isogonal conjugate of X(8684)
X(3808) = crossdifference of every pair of points on line X(6)X(983)

### X(3809) = X(2786)com[INVERSE(nT(bc, b2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2a2 + bc)(b2 + c2 + bc)

X(3809) lies on this line: {1, 2}

### X(3810) = X(525)com[INVERSE(nT(-bc, c2))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)(b + c - a)(b2 + c2 - bc)

X(3810) lies on this line: {30, 511}

X(3810) = isogonal conjugate of X(8685)

### X(3811) = X(3189)com(EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a3 + b3 + c3 - a2b - ab2 - a2c - ac2 + b2c + bc2 - 2abc)

X(3811) lies on these lines: {1, 2}, {3, 518}, {4, 2900}, {6, 3694}, {9, 943}, {12, 3419}, {21, 3681}, {33, 2901}, {35, 63}, {37, 2271}, {40, 758}, {46, 100}, {47, 1331}, {55, 72}, {56, 3555}, {58, 1792}, {65, 3689}, {84, 2801}, {101, 1973}, {158, 1897}, {169, 3684}, {210, 405}, {214, 1420}, {218, 3693}, {354, 474}, {392, 3303}, {404, 3338}, {443, 3475}, {500, 524}, {516, 1490}, {595, 3749}, {726, 990}, {908, 1479}, {912, 1158}, {942, 1376}, {954, 3059}, {956, 2646}, {960, 3295}, {993, 3601}, {1046, 3550}, {1066, 1818}, {1389, 3680}, {1612, 1723}, {1706, 3754}, {2136, 2802}, {2177, 2292}, {2321, 3553}, {2550, 3487}, {2551, 3488}, {2975, 3612}, {3243, 3333}, {3421, 3486}, {3697, 3711}

### X(3812) = X(950)com(4th EULER TRIANGLE)

Barycentrics    a(b3 + c3 - a2b - a2c - 3b2c - 3bc2 - 2abc) : :
X(3812) = 3*X(2) + X(65)

X(3812) lies on these lines: {1, 474}, {2, 65}, {6, 1722}, {7, 2551}, {8, 354}, {9, 3339}, {10, 141}, {21, 1155}, {29, 1888}, {37, 986}, {40, 1001}, {44, 1046}, {46, 405}, {56, 3306}, {57, 958}, {72, 1698}, {75, 3714}, {140, 517}, {171, 1104}, {226, 1329}, {241, 1254}, {377, 1837}, {392, 3624}, {404, 2646}, {442, 1737}, {452, 3474}, {475, 1905}, {595, 3246}, {758, 3634}, {899, 2650}, {908, 3649}, {938, 2550}, {956, 3338}, {1210, 2886}, {1738, 1834}, {1836, 2478}, {1858, 2476}, {1875, 1940}, {2295, 3290}, {2496, 3309}, {2802, 3636}, {3057, 3616}, {3452, 3671}, {3555, 3679}

X(3812) = centroid of ABCX(65)
X(3812) = Kosnita(X(65),X(2)) point
X(3812) = complement of X(960)
X(3812) = perspector of Gemini triangle 21 and cross-triangle of Gemini triangles 21 and 23

### X(3813) = X(2136)com(4th EULER TRIANGLE)

Barycentrics    (b + c - a)(b3 + c3 + a2b + a2c - b2c - bc2 - 4abc) : :

X(3813) lies on these lines: {1, 442}, {2, 3303}, {3, 528}, {4, 529}, {5, 519}, {8, 11}, {10, 496}, {12, 145}, {21, 3058}, {56, 3434}, {149, 2975}, {377, 3304}, {452, 497}, {495, 3244}, {499, 3035}, {518, 946}, {522, 596}, {535, 3627}, {943, 1001}, {956, 1479}, {1213, 3169}, {1376, 3086}, {1484, 2802}, {1698, 2136}, {2476, 3241}, {3158, 3624}, {3189, 3616}, {3679, 3680}, {3702, 3703}, {3704, 3705}

### X(3814) = X(2077)com(EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 - a2b2 - a2c2 + ab2c + abc2 - 2b2c2)

X(3814) = 2r*X(5) - R*X(10)
X(3814) = 3*X(2) - X(36)     (Peter Moses, April 2, 2012)

X(3814) lies on these lines: {2, 36}, {4, 2077}, {5, 10}, {11, 519}, {12, 1125}, {30, 3035}, {100, 3583}, {115, 1575}, {119, 214}, {121, 3259}, {381, 1376}, {403, 1861}, {404, 3585}, {427, 1878}, {442, 1155}, {484, 1698}, {495, 551}, {496, 3244}, {498, 2478}, {499, 3436}, {516, 1532}, {758, 908}, {958, 1656}, {1107, 1506}, {1621, 3584}, {1739, 3120}, {2392, 3142}, {2550, 3545}, {2551, 3090}, {3836, 4129}

X(3814) = inverse-in-nine-point-circle of X(10)
X(3814) = complement of X(36)

### X(3815) = X(574)com(5th EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 - 3a2b2 - 3a2c2 - 2b2c2

X(3815) lies on these lines: {2, 6}, {3, 2548}, {5, 39}, {11, 2276}, {12, 2275}, {25, 160}, {30, 574}, {32, 140}, {53, 232}, {187, 549}, {216, 1368}, {233, 1196}, {251, 2965}, {262, 1513}, {381, 2549}, {495, 1015}, {496, 1500}, {543, 3363}, {566, 858}, {594, 3705}, {631, 1285}, {1107, 1329}, {1575, 2886}, {1595, 3199}, {2207, 3541}, {2963, 3108}

X(3815) = crosspoint of X(2) and X(262)
X(3815) = crosssum of X(6) and X(182)
X(3815) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,6,230), (5,39,5254), (395,396,597)
X(3815) = X(6)-of-5th-Euler triangle
X(3815) = inverse-in-{circumcircle, nine-point circle}-inverter of X(352)
X(3815) = insimilicenter of nine-point and (1/2)-Moses circles; the exsimilicenter is X(5254)

### X(3816) = X(57)com(3rd EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b3 + c3 - a2b - a2c - b2c - bc2 + 4abc

X(3816) lies on these lines: {1, 1329}, {2, 11}, {5, 515}, {8, 1997}, {10, 496}, {12, 1388}, {38, 1647}, {56, 2478}, {142, 1538}, {226, 3660}, {354, 908}, {392, 1737}, {405, 499}, {442, 3586}, {474, 1479}, {495, 551}, {518, 3452}, {529, 999}, {958, 3086}, {960, 1210}, {978, 1834}, {982, 3756}, {1532, 3576}, {1836, 3306}, {3304, 3436}

### X(3817) = X(2)com(3rd EULER TRIANGLE)

Barycentrics    3b3 + 3c3 - a2b - a2c - 2ab2 - 2ac2 - 3b2c - 3bc2 + 4abc : :

In the plane of a triangle ABC, let I = X(1) and
L = line through midpoint of CA perpendicular to BI
L' = line through midpoint of AB perpendicular to CI
L'' = line through midpoint of AI perpendicular to BC
The lines L, L', L'' concur in a point, A'; define B' and C' cyclically. Ten X(3817) = X(2)-of-A'B'C'. Also, A'B'C' = complement of the excentral triangle, and A'B'C' = extraversion triangle of X(10). (Randy Hutson, September 14, 2016)

Let A'B'C' be the intouch triangle, and AaAbAc, BaBbBc, CaCbCc the A-, B-, and C-extouch triangles. Let Ab' = B'C'∩BcBa and Ac' = B'C'∩CaCb. Define Bc' and Ba', Ca', Cb' cyclically. Then X(3817) is the centroid of {Ab',Ac',Bc',Ba',Ca',Cb'} is X(3817). The six points lie on a common ellipse. (Randy Hutson, September 14, 2016)

Let Wa be the inverter of the B- and C-excircles, and define Wb and Wc cyclically; see X(5577). Let Ia be Wa-inverse of the incircle, and define Ib and Ic cyclically. The radical center of circles Ia, Ib, Ic is X(3817). (Randy Hutson, September 14, 2016)

Let A'B'C' be the circumcevian triangle of X(1). Let L1 be the Simson line of A', and define L2 and L3 cyclically. Let A'' = L2∩L3, and define B'' and C'' cyclically. Then X(3817) = X(2)-of-A''B''C''. (Angel Montesdeoca, June 28, 1017)

X(3817) is the centroid of AbAcBcBaCaCb in the construction of the perspeconic of the Ursa-minor and Ursa-major triangles. (Randy Hutson, June 27, 2018)

X(3817) lies on these lines: {1, 3091}, {2, 165}, {4, 1125}, {5, 10}, {11, 118}, {20, 3624}, {40, 3090}, {57, 1776}, {142, 1538}, {262, 726}, {355, 3244}, {381, 515}, {519, 3545}, {546, 1385}, {748, 1754}, {908, 3681}, {944, 3636}, {962, 1698}, {971, 3742}, {1210, 3671}, {1323, 2898}, {1482, 3625}, {1709, 3306}, {3057, 3614}, {3579, 3628}

X(3817) = complement of X(165)
X(3817) = centroid of 3rd Euler triangle
X(3817) = homothetic center of 3rd Euler triangle and extraversion triangle of X(10)

### X(3818) = X(1352)com(EULER TRIANGLE)

Barycentrics    b6 + c6 - a6 - b4c2 - b2c4 - 2a2b2c2 : :
Barycentrics    SA^2 (SB + SC) + SB SC (4 SA + 3 SB + 3 SC) : :

X(3818) lies on these lines: {2, 1495}, {3, 2916}, {4, 69}, {5, 182}, {6, 13}, {30, 141}, {98, 3407}, {125, 1995}, {147, 262}, {302, 383}, {303, 1080}, {343, 428}, {376, 3619}, {382, 1350}, {403, 1974}, {546, 576}, {575, 3091}, {1469, 3585}, {1539, 2781}, {1619, 1853}, {3056, 3583}, {3060, 3410}, {3543, 3620}, {3545, 3618}

X(3818) = intersection of Fermat axes of ABC and 1st Brocard triangle
X(3818) = inverse-in-Kiepert-hyperbola of X(5309)
X(3818) = X(7)-of-Ehrmann-vertex-triangle if ABC is acute
X(3818) = X(6)-of-Ehrmann-mid-triangle
X(3818) = {X(13),X(14)}-harmonic conjugate of X(5309)
X(3818) = perspector of Ehrmann vertex-triangle and Ehrmann mid-triangle
X(3818) = intersection, other than X(4), of the Brocard circles of the 1st and 2nd Ehrmann circumscribing triangles

### X(3819) = X(549)com(2nd EULER TRIANGLE)

Barycentrics    a2(b4 + c4 - a2b2 - a2c2 + 4b2c2) : :

Let A'B'C' be the medial triangle. Let Ab and Ac be the circumcircle intercepts of line B'C'. Define Bc and Ca cyclically, and define Ba and Cb cyclically. X(3819) is the centroid of AbAcBcBaCaCb. (Randy Hutson, July 31 2018)

X(3819) lies on these lines: {2, 51}, {3, 64}, {25, 3098}, {39, 1613}, {52, 3526}, {122, 129}, {140, 389}, {141, 1368}, {182, 394}, {185, 3523}, {187, 1915}, {210, 2810}, {474, 970}, {575, 1993}, {674, 3742}, {960, 2390}, {982, 3688}, {984, 1401}, {1194, 3231}, {1196, 3094}, {1843, 3619}, {3218, 3690}, {3533, 3567}

X(3819) = complement of X(51)
X(3819) = X(2) of polar triangle of complement of polar circle

### X(3820) = X(200)com(4th EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 - a2b2 - a2c2 + 4ab2c + 4abc2 + 2b2c2

X(3820) lies on these lines: {2, 495}, {3, 1603}, {5, 10}, {8, 496}, {9, 119}, {11, 3679}, {12, 57}, {30, 1376}, {116, 121}, {140, 958}, {210, 1737}, {329, 442}, {355, 936}, {381, 2550}, {474, 3436}, {549, 993}, {908, 3753}, {952, 997}, {1484, 3036}, {1596, 1861}, {2885, 3454}

### X(3821) = X(3056)com(4th EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b3 + c3 + a2b + a2c + ab2 + ac2

X(3821) lies on these lines: {1, 2896}, {2, 846}, {3, 142}, {10, 75}, {45, 3634}, {86, 1326}, {141, 740}, {182, 2792}, {226, 1403}, {519, 599}, {579, 1761}, {752, 1386}, {1352, 2784}, {1698, 3729}, {2085, 3670}, {2092, 3454}, {2887, 3666}, {3094, 3735}

X(3821) = X(10) of 1st Brocard triangle

### X(3822) = X(2099)com(3rd EULER TRIANGLE)

Barycentrics   (b + c)(b3 + c3 - a2b - a2c - b2c - bc2 - abc) : :

Suppose that ABC is an acute triangle. Let LA be the circle that is externally tangent to the nine-point circle and to the sidelines AB and AC. Define LB and LC cyclically. The three circles are here named the Odehnal tritangent circles. Their radical center is X(3822). The center of the Apollonian circle of LA, LB, LC is X(6167). (Boris Odehnal, "A Triad of Tritangent Circles, Journal for Geometry and Graphics 18 (2014) no. 1, 61-71)

X(3822) lies on these lines: {1, 2476}, {2, 36}, {5, 515}, {10, 12}, {11, 551}, {21, 3585}, {35, 2475}, {37, 115}, {63, 1698}, {100, 3584}, {116, 119}, {377, 498}, {381, 1001}, {495, 519}, {496, 3636}, {1329, 3634}, {1621, 3583}, {2901, 3178}

### X(3823) = X(3717)com(4th EULER TRIANGLE)

Barycentrics    2b3 + 2c3 + a2b + a2c - ab2 - ac2 - 4abc : :

X(3823) lies on these lines: {2, 1279}, {10, 141}, {44, 966}, {238, 1698}, {536, 1738}, {1086, 3717}, {2887, 3740}, {3246, 3634}

### X(3824) = X(3671)com(3rd EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a + 2b + 2c)(b3 + c3 - a2b - a2c - 2abc - b2c - bc2)

X(3824) lies on these lines: {5, 142}, {30, 1125}, {79, 3683}, {442, 942}, {527, 3634}, {1698, 3715}, {3454, 3739}

### X(3825) = X(56)com(3rd EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 - a2b2 - a2c2 + 2a2bc + ab2c + abc2 + 2b2c2

X(3825) lies on these lines: {2, 35}, {5, 515}, {10, 11}, {12, 551}, {37, 1506}, {56, 535}, {404, 3583}, {495, 3636}, {496, 519}, {499, 993}, {758, 1210}, {946, 3754}, {1001, 1656}, {2476, 3624}, {2886, 3634}, {2975, 3582}, {3452, 3678}

### X(3826) = X(3243)com(3rd EULER TRIANGLE)

Barycentrics    b3 + c3 - ab2 - ac2 - b2c - bc2 - 4abc : :

X(3826) lies on these lines: {2, 11}, {5, 516}, {7, 12}, {9, 46}, {10, 141}, {37, 1738}, {226, 3740}, {427, 1890}, {443, 958}, {498, 954}, {984, 1086}, {1269, 3701}, {1386, 3008}, {1827, 1861}, {1836, 3305}, {3243, 3679}

X(3826) = complement of X(1001)
X(3826) = X(6)-of-4th-Euler-triangle
X(3826) = inverse-in-Feuerbach-hyperbola of X(3058)
X(3826) = centroid of trapezoid X(1)X(7)X(8)X(9)

### X(3827) = X(518)com(2nd EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b5 + c5 - a4b - a4c - b4c - bc4 + 2a3bc)

X(3827) lies on these lines: {1, 159}, {6, 19}, {30, 511}, {66, 72}, {141, 960}, {154, 354}, {197, 1763}, {206, 942}, {210, 1853}, {1214, 3185}, {1279, 2195}, {1633, 3100}, {1824, 1836}, {1828, 1837}, {1843, 1858}, {1854, 3057}

X(3827) = crossdifference of every pair of points on line X(6)X(521)

### X(3828) = X(210)com(4th EULER TRIANGLE)

Trilinears    3 r - 10 R sin B sin C : :
Barycentrics    2a+5b+5c : :
X(3828) = X(1) - 5 X(2) = X(8) - 7 X(10)

X(3828) lies on these lines: {1, 2}, {12, 553}, {40, 3545}, {115, 121}, {165, 3543}, {375, 2392}, {381, 516}, {515, 549}, {517, 547}, {537, 3739}, {756, 1739}, {758, 3740}, {903, 1268}, {946, 3654}, {1656, 3656}, {3526, 3653}

X(3828) = midpoint of X(2) and X(10)
X(3828) = complement of X(551)
X(3828) = {X(1),X(2)}-harmonic conjugate of X(19883)
X(3828) = centroid of mid-triangle of Gemini triangles 19 and 20

### X(3829) = X(3158)com(3rd EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3b3 + 3c3 - 3ab2 - 3ac2 + 4abc - 3b2c - 3bc2

X(3829) lies on these lines: {2, 11}, {5, 519}, {12, 3241}, {145, 3614}, {381, 529}, {496, 551}, {1329, 3679}

### X(3830) = X(3543)com(EULER TRIANGLE)

Barycentrics    5a4 - 4b4 - 4c4 - a2b2 - a2c2 + 8b2c2 : :
X(3830) = 4X(2) - 3X(3) = X(2) - 3X(4)

As a point on the Euler line, X(3830) has Shinagawa coefficients (1, -9).

Let T=A'B'C' be a triangle directly similar to ABC and inscribed in the circumcircle. As T varies, the isogonal conjugate of the trilinear product A'*B'*C' traces the Yff hyperbola, centered at X(381) with vertices at X(2) and X(4), and foci at X(3) and X(3830). (Randy Hutson, June 27, 2018)

X(3830) lies on these lines: {2, 3}, {115, 1384}, {265, 541}, {394, 1531}, {399, 1539}, {515, 3656}, {516, 3654}, {542, 1351}, {946, 3655}, {999, 3583}, {1159, 1836}, {1478, 3058}, {1853, 2777}, {3295, 3585}, {3521, 3527}

X(3830) = anticomplement of X(8703)

X(3830) = midpoint of X(381) and X(382)
X(3830) = reflection of X(3) in X(381)
X(3830) = {X(2),X(3)}-harmonic conjugate of X(15701)
X(3830) = {X(2),X(4)}-harmonic conjugate of X(3845)
X(3830) = {X(2043),X(2044)}-harmonic conjugate of X(140)
X(3830) = Ehrmann-mid-to-ABC similarity image of X(2)

### X(3831) = X(3702)com(4th EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2b2 + a2c2 + ab3 + ac3 + b3c + bc3 + 2b2c2

X(3831) lies on these lines: {1, 2}, {5, 2887}, {38, 3701}, {141, 1329}, {312, 986}, {726, 1089}, {750, 964}, {942, 1215}, {992, 3686}, {1861, 3144}, {2277, 2321}, {2392, 3142}, {3663, 3760}, {3714, 3752}

### X(3832) = X(3090)com(EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3a4 - 5b4 - 5c4 + 2a2b2 + 2a2c2 + 10b2c2

As a point on the Euler line, X(3832) has Shinagawa coefficients (1, 4).

X(3832) lies on these lines: {2, 3}, {6, 1131}, {8, 1699}, {11, 3600}, {12, 390}, {64, 3066}, {145, 946}, {355, 3621}, {489, 3595}, {490, 3593}, {515, 3622}, {962, 3617}, {3085, 3583}, {3086, 3585}

X(3832) = {X(1131),X(1132)}-harmonic conjugate of X(6)

### X(3833) = X(3058)com(4th EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b3 + c3 - a2b - a2c - 4b2c - 4bc2 - 4abc)

X(3833) lies on these lines: {2, 758}, {10, 354}, {116, 119}, {140, 517}, {519, 3742}, {547, 2771}, {551, 2802}, {942, 3634}, {993, 3306}, {1698, 3681}, {1739, 3720}, {2836, 3589}, {3244, 3698}, {3336, 3647}

X(3833) = complement of X(10176)

### X(3834) = X(1266)com(3rd EULER TRIANGLE)

Barycentrics   ab + ac + 2bc - 2b2 - 2c2 : :

X(3834) lies on these lines: {2, 44}, {10, 141}, {37, 3662}, {238, 3624}, {244, 2228}, {513, 3716}, {524, 3008}, {536, 1086}, {545, 2325}, {752, 1125}, {1279, 3616}, {2887, 3742}, {3589, 3664}, {3631, 3686}

X(3834) = complement of X(44)

### X(3835) = X(693)com(3rd EULER TRIANGLE)

Barycentrics    (b - c)(ab + ac - bc) : :

Let A'B'C' be the Aquila triangle. Let A" be the trilinear product B'*C', and define B", C" cyclically. A"B"C" is homothetic to ABC at X(3835) and to A'B'C' at X(10). (Randy Hutson, June 27, 2018)

X(3835) lies on these lines: {2, 649}, {116, 3259}, {226, 3676}, {512, 625}, {513, 3716}, {514, 661}, {522, 1491}, {650, 812}, {788, 3741}, {802, 3709}, {824, 3004}, {1538, 3309}, {1848, 3064}, {2254, 3667}

X(3835) = isotomic conjugate of X(4598)
X(3835) = complement of X(649)
X(3835) = trilinear product of vertices of the Aquila triangle
X(3835) = X(1)-of-cross-triangle-of Aquila-and-anti-Aquila-triangles
X(3835) = center of the perspeconic of the Ursa-minor and Ursa-major triangles

### X(3836) = X(3685)com(4th EULER TRIANGLE)

Barycentrics    b3 + c3 - 2abc

X(3836) lies on these lines: {2,31}, {5,15310}, {8,17232}, {9,4655}, {10,141}, {11,4871}, {12,1463}, {37,3821}, {42,18139}, {43,18134}, {44,1213}, {57,4438}, {75,3773}, {120,17793}, {190,17767}, {239,17772}, {244,3006}, {305,18067}, {312,17889}, {320,1698}, {334,1921}, {344,24248}, {346,7613}, {442,3831}, {513,3814}, {516,4432}, {519,4864}, {536,6541}, {537,3717}, {614,4865}, {726,1086}, {740,1738}, {742,25357}, {756,17184}, {899,3936}, {902,24542}, {908,4892}, {940,25453}, {946,21626}, {960,25108}, {984,3662}, {1001,4660}, {1089,1269}, {1125,1279}, {1193,17674}, {1215,5249}, {1326,25536}, {1329,20258}, {1376,3771}, {1386,17356}, {1836,4011}, {1961,19786}, {2140,12263}, {2886,3840}, {3008,4974}, {3011,4434}, {3120,4358}, {3123,22220}, {3246,19862}, {3290,4071}, {3305,4703}, {3416,16825}, {3452,4138}, {3663,4078}, {3666,24169}, {3670,21035}, {3685,17266}, {3696,17231}, {3703,24165}, {3705,17063}, {3720,4972}, {3726,4119}, {3741,3925}, {3751,17298}, {3782,3971}, {3842,4357}, {3909,20962}, {3923,5880}, {3944,18743}, {3993,17243}, {4009,21093}, {4136,20271}, {4167,21951}, {4359,15523}, {4362,24789}, {4407,17227}, {4417,16569}, {4422,17768}, {4431,4535}, {4446,24046}, {4527,17233}, {4649,17300}, {4663,17376}, {4672,17353}, {4676,17341}, {4697,5294}, {4698,25354}, {4710,18040}, {4716,6542}, {4743,17241}, {4980,6535}, {5087,11814}, {5205,17719}, {5220,7232}, {5248,16299}, {5263,17283}, {5268,25527}, {5695,17267}, {5852,7238}, {5853,17059}, {5988,9478}, {6685,17056}, {7295,16048}, {7800,25500}, {8287,20546}, {9470,20341}, {9780,17238}, {12047,25079}, {12782,24190}, {15481,17345}, {16415,25440}, {16468,17352}, {16830,17291}, {17047,24206}, {17237,25352}, {17260,24697}, {17263,24723}, {17284,24693}, {17289,24342}, {17307,19856}, {17357,24295}, {17748,20227}, {19557,19559}, {19877,20072}, {20337,20339}, {20340,20343}, {20432,20703}, {20491,20532}, {20530,20531}, {20947,24731}, {24176,24180}, {24358,24699}

### X(3837) = X(2254)com(3rd EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)(ab2 + ac2 - b2c - bc2)

X(3837) lies on these lines: {2, 659}, {5, 2826}, {10, 891}, {11, 244}, {120, 2977}, {325, 523}, {334, 876}, {513, 3716}, {661, 1639}, {814, 905}, {946, 2821}, {1125, 1960}, {1577, 2530}

X(3837) = isotomic conjugate of X(8709)

### X(3838) = X(226)com(3rd EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2b3 + 2c3 - a2b - a2c - ab2 - ac2 - 2b2c - 2bc2

X(3838) lies on these lines: {2, 1155}, {11, 3742}, {30, 1125}, {65, 2476}, {142, 1538}, {149, 3748}, {226, 518}, {442, 960}, {908, 3740}, {1001, 1699}, {1376, 3256}, {2475, 2646}, {3120, 3666}

### X(3839) = X(3545)com(EULER TRIANGLE)

Barycentrics    5a4 - 7b4 - 7c4 + 2a2b2 + 2a2c2 + 14 b2c2 : :
X(3839) = 7 X(2) - 4 X(3)

As a point on the Euler line, X(3839) has Shinagawa coefficients (1, 6).

X(3839) lies on these lines: {2, 3}, {145, 3656}, {147, 671}, {390, 3583}, {519, 1699}, {598, 3424}, {946, 3241}, {962, 3679}, {1131, 1328}, {1132, 1327}, {3585, 3600}, {3622, 3655}

X(3839) = {X(2),X(3)}-harmonic conjugate of X(15721)
X(3839) = trisector nearest X(4) of segment X(2)X(4)
X(3839) = endo homothetic center of Ehrmann side-triangle and submedial triangle; the homothetic center is X(5055)

### X(3840) = X(3210)com(3rd EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ab2 + ac2 + b2c + bc2 - 2abc

X(3840) lies on these lines: {1, 2}, {11, 2887}, {226, 1401}, {244, 321}, {312, 726}, {350, 3663}, {354, 1215}, {740, 3752}, {748, 1150}, {1575, 2321}, {2810, 3038}, {3551, 3662}

### X(3841) = X(3210)com(4th EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b3 + c3 - a2b - a2c - b2c - bc2 - 3abc)

X(3841) lies on these lines: {2, 35}, {5, 516}, {10, 12}, {79, 3219}, {377, 993}, {484, 1698}, {495, 3626}, {496, 1125}, {535, 958}, {626, 3739}, {759, 833}, {1770, 3647}

### X(3842) = X(3688)com(4th EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(a2 + 2ab + 2ac + bc)

X(3842) lies on these lines: {1, 872}, {2, 38}, {10, 37}, {75, 1089}, {86, 1757}, {333, 1961}, {518, 1125}, {726, 3634}, {1237, 1921}, {2664, 3736}, {2667, 3293}

### X(3843) = X(3091)com(EULER TRIANGLE)

Barycentrics   3a4 - 4b4 - 4c4 + a2b2 + a2c2 + 8b2c2 : :

As a point on the Euler line, X(3843) has Shinagawa coefficients (1, 7).

X(3843) lies on these lines: {2, 3}, {68, 3531}, {265, 3527}, {355, 3625}, {946, 3635}, {999, 3585}, {1159, 1837}, {1352, 3630}, {1482, 1699}, {3295, 3583}, {3426, 3521}

X(3843) = homothetic center of Ehrmann vertex-triangle and anti-incircle-circles triangle
X(3843) = homothetic center of Ehrmann mid-triangle and X3-ABC reflections triangle
X(3843) = endo-homothetic center of Ehrmann mid-triangle and anti-Euler triangle; the homothetic center is X(3091)
X(3843) = {X(2043),X(2044)}-harmonic conjugate of X(547)

### X(3844) = X(2321)com(4th EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2b3 + 2c3 + a2b + a2c + ab2 + ac2 + 2b2c + 2bc2

X(3844) lies on these lines: {2, 1386}, {6, 1698}, {8, 3619}, {10, 141}, {120, 125}, {594, 1738}, {599, 3751}, {3242, 3679}, {3589, 3634}, {3661, 3696}

### X(3845) = X(381)com(EULER TRIANGLE)

Barycentrics    4a4 - 5b4 - 5c4 + a2b2 + a2c2 + 10b2c2 : :
X(3845) = X(2) + 3 X(4)

As a point on the Euler line, X(3845) has Shinagawa coefficients (1, 9).

X(3845) is the point P which divides segment X(4)X(5) in the ratio PX(4)/PX(5) = -1/2. (Randy Hutson, June 27, 2018)

X(3845) lies on these lines: {2, 3}, {6, 1327}, {343, 1531}, {399, 1994}, {495, 3058}, {496, 3585}, {541, 1539}, {542, 1353}, {946, 1483}, {952, 1699}

X(3845) = {X(2),X(4)}-harmonic conjugate of X(3830)
X(3845) = {X(13595),X(13596)}-harmonic conjugate of X(3)
X(3845) = Ehrmann-side-to-orthic similarity image of X(2)
X(3845) = Johnson-to-Ehrmann-mid similarity image of X(381)
X(3845) = trisector nearest X(4) of segment X(4)X(5)
X(3845) = X(2)-of-Ehrmann-mid-triangle
X(3845) = X(10246)-of-orthic-triangle if ABC is acute

### X(3846) = X(1999)com(3rd EULER TRIANGLE)

Barycentrics    b3 + c3 + 2abc : :

X(3846) lies on these lines: {2,31}, {5,10}, {9,4438}, {11,1211}, {42,5741}, {43,4085}, {57,4655}, {63,4703}, {75,3944}, {114,124}, {141,3816}, {142,4138}, {226,24325}, {244,17184}, {312,3773}, {325,4357}, {612,4865}, {726,4415}, {740,3687}, {756,3006}, {899,4972}, {908,1215}, {978,16062}, {984,3705}, {1001,3771}, {1100,10026}, {1104,1125}, {1193,5051}, {1203,25441}, {1376,4660}, {1836,3980}, {1921,7018}, {1999,17772}, {3120,4359}, {3178,6051}, {3218,4683}, {3266,21415}, {3662,17063}, {3666,4425}, {3670,14815}, {3702,20653}, {3703,3971}, {3706,21085}, {3717,4096}, {3720,3936}, {3727,4167}, {3739,3838}, {3752,3821}, {3757,17719}, {3772,16825}, {3782,24165}, {3831,4187}, {3834,3848}, {3844,11814}, {3923,24703}, {3925,5241}, {3931,17748}, {3961,4514}, {3966,4362}, {4011,4679}, {4023,4685}, {4026,6685}, {4038,17778}, {4104,4847}, {4199,21321}, {4358,15523}, {4383,25453}, {4384,17064}, {4389,17591}, {4418,5057}, {4429,16569}, {4647,6042}, {4854,4970}, {4892,5249}, {5015,5293}, {5211,17598}, {5259,25645}, {5272,25527}, {5278,24892}, {5739,11269}, {5814,17733}, {5955,12699}, {7741,10479}, {7988,18229}, {8167,16846}, {9284,16584}, {10453,24217}, {11680,21242}, {15507,18235}, {15825,19863}, {16610,24169}, {16739,17203}, {16817,24161}, {17202,23659}, {17234,25502}, {17274,18193}, {17355,17747}, {17357,21249}, {17596,24723}, {17605,25385}, {17889,19804}, {19543,25440}, {19563,20333}, {21856,25066}

X(3846) = complement of X(171)

### X(3847) = X(1420)com(3rd EULER TRIANGLE)

Barycentrics   (b + c - a)(3b3 + 3c3 + 3ab2 + 3ac2 - 3b2c - 3bc2 - 4abc) : :

X(3847) lies on these lines: {2,5217}, {3,6667}, {4,6691}, {5,515}, {8,11}, {10,3829}, {12,3622}, {55,6931}, {56,5187}, {496,3244}, {528,9669}, {529,3086}, {550,6681}, {958,6919}, {1001,3090}, {1210,5087}, {1479,3035}, {1656,6690}, {1698,1706}, {2478,4999}, {3614,3616}, {3628,5248}, {3812,3817}, {3820,4691}, {3838,9843}, {3878,6797}, {3913,5274}, {4189,7294}, {4423,6933}, {5046,5303}, {5057,7098}, {5690,6702}, {5842,6959}, {5880,6831}, {6253,6979}, {6856,8167}, {6882,7681}, {6971,7680}

### X(3848) = X(553)com(3rd EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 - ab - ac - 8bc)

X(3848) lies on these lines: {2, 210}, {140, 517}, {142, 1538}, {165, 1001}, {171, 3246}, {960, 3624}, {3149, 3576}, {3622, 3698}

### X(3849) = X(524)com(5th EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 4a4 - 2b4 - 2c4 - a2b2 - a2c2 + 2b2c2

X(3849) lies on these lines: {2, 187}, {30, 511}, {325, 2482}, {381, 2080}, {385, 671}, {597, 2030}, {599, 3734}, {1992, 2549}

X(3849) = isogonal conjugate of X(6323)
X(3849) = infinite point of Brocard axis of the 3rd pedal triangle of X(6)

### X(3850) = X(140)com(EULER TRIANGLE)

Barycentrics    2a4 - 5b4 - 5c4 + 3a2b2 + 3a2c2 + 10b2c2 : :
X(3850) = X(4) + 3*X(5)

As a point on the Euler line, X(3850) has Shinagawa coefficients (3, 7).

X(3850) lies on these lines: {2, 3}, {355, 3633}, {399, 1199}, {946, 3625}, {952, 3635}, {1327, 3594}, {1328, 3592}, {3583, 3614}

X(3850) = {X(4),X(5)}-harmonic conjugate of X(140)
X(3850) = X(13624)-of-orthic-triangle if ABC is acute

### X(3851) = X(3523)com(EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4 - 4b4 - 4c4 + 3a2b2 + 3a2c2 + 8b2c2

As a point on the Euler line, X(3851) has Shinagawa coefficients (3, 5)

X(3851) lies on these lines: {2, 3}, {355, 3244}, {946, 3626}, {1352, 3629}, {1479, 3614}, {1482, 3632}, {3519, 3527}

X(3851) = inverse-in-orthocentroidal-circle of X(550)
X(3851) = homothetic center of Euler triangle and mid-triangle of medial and anticomplementary triangles
X(3851) = homothetic center of 2nd isogonal triangle of X(4) and mid-triangle of orthic and circumorthic triangles
X(3851) = homothetic center of X(140)-altimedial and X(4)-anti-altimedial triangles

### X(3852) = X(732)com(2nd EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4(b8 + c8 - a4b4 - a4c4)

X(3852) lies on these lines: {6, 695}, {30, 511}, {32, 206}, {66, 315}, {141, 3491}, {159, 3499}

### X(3853) = X(3627)com(EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 6a4 - 5b4 - 5c4 - a2b2 - a2c2 + 10b2c2
X(3853) = 3*X(4) - X(5)

As a point on the Euler line, X(3853) has Shinagawa coefficients (1, -9).

X(3853) lies on these lines: {2, 3}, {1327, 3592}, {1328, 3594}

X(3853) = {X(4),X(5)}-harmonic conjugate of X(3861)

### X(3854) = X(3533)com(EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 5a4 - 11b4 - 11c4 + 6a2b2 + 6a2c2 + 22b2c2

As a point on the Euler line, X(3854) has Shinagawa coefficients (3, 8).

X(3854) lies on these lines: {2, 3}, {946, 3621}, {1699, 3617}

### X(3855) = X(3525)com(EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3a4 - 7b4 - 7c4 + 4a2b2 + 4a2c2 + 14b2c2

As a point on the Euler line, X(3855) has Shinagawa coefficients (2, 5).

X(3855) lies on these lines: {2, 3}, {944, 3636}, {946, 3632}

### X(3856) = X(3628)com(EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 6a4 - 11b4 - 11c4 + 5a2b2 + 5a2c2 + 22b2c2

As a point on the Euler line, X(3856) has Shinagawa coefficients (5, 17).

X(3856) lies on this line: {2, 3}

### X(3857) = X(3526)com(EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 4a4 - 9b4 - 9c4 + 5a2b2 + 5a2c2 + 18b2c2

As a point on the Euler line, X(3857) has Shinagawa coefficients (5, 13)

X(3857) lies on this line: {2, 3}

### X(3858) = X(1656)com(EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 4a4 - 7b4 - 7c4 + 3a2b2 + 3a2c2 + 14b2c2

As a point on the Euler line, X(3858) has Shinagawa coefficients (3, 11)

X(3858) lies on this line: {2, 3}

### X(3859) = X(632)com(EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 6a4 - 13b4 - 13c4 + 7a2b2 + 7a2c2 + 26b2c2

As a point on the Euler line, X(3859) has Shinagawa coefficients (7, 19)

X(3859) lies on this line: {2, 3}

### X(3860) = X(547)com(EULER TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 10a4 - 17b4 - 17c4 + 7a2b2 + 7a2c2 + 34b2c2

As a point on the Euler line, X(3860) has Shinagawa coefficients (7, 27)

X(3860) lies on this line: {2, 3}

### X(3861) = X(546)com(EULER TRIANGLE)

Barycentrics    6a4 - 7b4 - 7c4 + a2b2 + a2c2 + 14b2c2 : :
X(3861) = 3*X(4) + X(5)

As a point on the Euler line, X(3861) has Shinagawa coefficients (1, 13).

X(3861) lies on this line: {2, 3}

X(3861) = {X(4),X(5)}-harmonic conjugate of X(3853)

Inverse Triangles and More Combos

The Euler triangle and the 2nd, 3rd, 4th, and 5th Euler triangles are discussed in the preamble to X(3758), along with eight other central triangles using the notation T(f(a,b,c), g(b,c,a)). Recall that the inverse of a normalized central triangle T, denoted by Inverse(nT) or Inverse(n(T)), is also a normalized central triangle. Barycentrics for the inverse of each of the 13 triangles and properties of these triangles are given (Peter Moses, December 2011) as follows:

Inverse(n(Euler triangle)) = T(f(a,b,c), g(b,c,a)), where
f(a,b,c) = (b2 - c2)2 + a2(3a2 - 4b2 - 4c2)
g(b,c,a) = b4 - (c2 - a2)2

Inverse(n(2nd Euler triangle)) = T(f(a,b,c), g(b,c,a)), where
f(a,b,c) = a2(a2 - b2 - c2)(a6 + b6 + c6 - a4b2 - a4c2 - a2b4 - a2c4 - 2a2b2c2 - b4c2 - b2c4)
g(b,c,a) = b2(b2 - c2 - a2(a6 + b6 - c6 - a4b2 - 3a4c2 - a2b4 + 3a2c4 + 2a2b2c2 - 3b4c2 + 3b2c4)

Inverse(n(3rd Euler triangle)) = T(f(a,b,c), g(b,c,a)), where
f(a,b,c) = a(b2 + c2 - ab - ac + bc)
g(b,c,a) = b(a2 - ab - ac + bc - c2)

Inverse(n(4th Euler triangle)) = T(f(a,b,c), g(b,c,a)), where
f(a,b,c) = a(a - b - c)(b2 + c2 + ab + ac)
g(b,c,a) = b(-a + b + c)(a2 + ab - ac - bc - c2)

Inverse(n(5th Euler triangle)) = T(f(a,b,c), g(b,c,a)), where
f(a,b,c) = (a2 - 2b2 - 2c2)(3a2 + b2 + c2)
g(b,c,a) = -(a2 - b2 - c2)(2a2 - b2 + 2c2)

Inverse(n(T(bc,b2)) = T(f(a,b,c), g(b,c,a)), where
f(a,b,c) = -bc(a2 - bc)(b2 + c2 + bc)
g(b,c,a) = b2(ab - c2)(a2 + c2 + ac)

Inverse(n(T(-bc,b2)) = T(f(a,b,c), g(b,c,a)), where
f(a,b,c) = bc(a2 - bc)(b2 + c2 - bc)
g(b,c,a) = b2(ab + c2)(a2 + c2 - ac)

Inverse(n(T(bc,c2)) = T(f(a,b,c), g(b,c,a)), where
f(a,b,c) = -a(a2 - bc)(b2 + c2 + bc)
g(b,c,a) = (ab - c2)(a2 + c2 + ac)

Inverse(n(T(-bc,c2)) = T(f(a,b,c), g(b,c,a)), where
f(a,b,c) = -a(a2 - bc)(b2 + c2 - bc)
g(b,c,a) = (ab + c2)(a2 + c2 - ac)

Inverse(n(T(a2,a2 - b2)) = T(f(a,b,c), g(b,c,a)), where
f(a,b,c) = (3a2 - b2 - c2)(b4 + c4 - b2c2)
g(b,c,a) = (a2 - 3b2 + c2)(a2b2 - 2b2c2 + c4)

Inverse(n(T(a2,a2 - c2)) = T(f(a,b,c), g(b,c,a)), where
f(a,b,c) = a2(b2 + c2 - 3a2)
g(b,c,a) =(a2 - c2)(a2 - 3b2 + c2)

Inverse(n(T(a2,a2 + b2)) = T(f(a,b,c), g(b,c,a)), where
f(a,b,c) = -(3a2 + b2 + c2)(b4 + c4 + b2c2)
g(b,c,a) = (a2 + 3b2 + c2)(a2b2 + c4)

Inverse(n(T(a2,a2 + c2)) = T(f(a,b,c), g(b,c,a)), where
f(a,b,c) = -a2(a2 + b2 + c2)(3a2 + b2 + c2)
g(b,c,a) = (a2 + c2)(a2 + b2 - c2)(a2 + 3b2 + c2)

For present purposes, label these last triangles as IT1, IT2, IT3, IT4, IT5, IT6, IT7, IT8; then all except IT5 and IT6 are perspective to the reference triangle ABC; perspectors are described at X(3862)-X(3866). Label the corresponding original triangles T1, T2, T3, T4, T5, T6, T7, T8; then the following pairs are perspective: (T1, IT1), (T2, IT2), (T3,IT3), (T4, IT4), (T7, IT7), (T8,IT8); coordinates for the perspectors are long and omitted.

The triangles IT1 and IT2 are triply perspective to ABC (as are T1 and T3). Order-label IT1 as A'B'C'. The three perspectivities are then given by

AA'∩BB'∩CC' = X(3862)
AB'∩BC'∩CA' = f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ab(a2 - bc)(c2 - ab)(b2 + bc + c2)
AC'∩BA'∩CB' = f(a,c,b) : f(b,a,c) : f(c,b,a)

Note that the second two perspectors are a bicentric pair (not triangle centers; see TCCT, page 47); likewise for the other triple perspectivity, with IT3 order-labeled as A'B'C':

AA'∩BB'∩CC' = X(3864)
AB'∩BC'∩CA' = g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a(a2 - bc)(c2 - ab)(b2 + bc + c2)
AC'∩BA'∩CB' = g(a,c,b) : g(b,a,c) : g(c,b,a)

### X(3862) =  PERSPECTOR OF ABC AND T(bc, b2)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b2 - ac)(c2 - ab)(b2 + bc + c2)

If you have The Geometer's Sketchpad, you can view X(3862).

X(3862) lies on these lines: {6,292}, {37,256}, {75,141}, {291,518}, {660,2235}, {753,813}, {984,3094}, {1333,2311}, {1755,2076}, {2276,3116}

{X(694),X(1581)}-harmonic conjugate of X(3863)

### X(3863) =  PERSPECTOR OF ABC AND T(-bc, b2)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b2 + ac)(c2 + ab)(b2 - bc + c2)

X(3863) lies on these lines: {6,893}, {37,256}, {75,257}, {904,1964}, {1178,1333}, {3056,3116}, {3061,3094}

X(3863) = {X(694),X(1581)}-harmonic conjugate of X(3862)

### X(3864) =  PERSPECTOR OF ABC AND T(bc, c2)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 - ac)(c2 - ab)(b2 + bc + c2)

X(3864) lies on these lines: {1,39}, {10,257}, {76,334}, {295,3497}, {335,726}, {511,1757}, {761,813}, {984,3094}

X(3864) = {X(1581),X(1916)}-harmonic conjugate of X(3865)

### X(3865) =  PERSPECTOR OF ABC AND T(-bc, c2)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + ac)(c2 + ab)(b2 - bc + c2)

X(3865) lies on these lines: {1,256}, {10,257}, {39,893}, {904,995}, {986,3095}, {3061,3094}

X(3865) = {X(1581),X(1916)}-harmonic conjugate of X(3864)

### X(3866) =  PERSPECTOR OF ABC AND T(a2, a2 + b2)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b4 + a2c2)(c4 + a2b2)(3a2 + b2 + c2)

X(3866) lies on these lines: {69,194}, {83,419}

### X(3867) =  PERSPECTOR OF ABC AND T(a2, a2 + c2)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 + c2)(a2 + b2 - c2)(a2 - b2 + c2)/(3a2 + b2 + c2)

X(3867) lies on these lines: {4,6}, {25,3589}, {141,427}, {428,597}, {511,1595}, {1350,3088}

### X(3868) = X(3)com[INVERSE(n(3rd EULER TRIANGLE))]

Barycentrics    a(b3 + c3 - a2b - a2c - abc)

X(3868) lies on these lines: {1, 21}, {2, 72}, {3, 3218}, {4, 912}, {6, 977}, {7, 8}, {10, 3681}, {20, 145}, {27, 3187}, {34, 651}, {41, 3509}, {42, 986}, {46, 100}, {56, 1259}, {57, 78}, {84, 1320}, {101, 1729}, {144, 452}, {146, 149}, {171, 976}, {193, 1829}, {200, 3339}, {226, 2476}, {239, 379}, {244, 978}, {286, 2997}, {326, 1014}, {329, 938}, {354, 960}, {355, 2888}, {386, 3670}, {392, 3622}, {405, 3219}, {412, 1897}, {497, 1858}, {527, 950}, {579, 2198}, {580, 1331}, {596, 994}, {664, 3188}, {894, 964}, {908, 1210}, {936, 3306}, {971, 3146}, {982, 1193}, {997, 3338}, {1012, 1482}, {1445, 1467}, {1475, 3061}, {1490, 1998}, {1697, 3243}, {1698, 3678}, {1870, 1993}, {2095, 3149}, {2176, 3726}, {2280, 3496}, {2475, 2894}, {2802, 3633}, {2886, 3649}, {3057, 3241}, {3189, 3474}, {3617, 3753}, {3679, 3754}

X(3868) = isogonal conjugate of X(2218)
X(3868) = isotomic conjugate of X(2997)
X(3868) = anticomplement of X(72)
X(3868) = perspector of Conway triangle and Gemini triangle 29
X(3868) = {X(1),X(63)}-harmonic conjugate of X(21)
X(3868) = {X(7),X(8)}-harmonic conjugate of X(377)
X(3868) = X(4)com[Inverse(n(4th Euler triangle))]

### X(3869) = X(3)com[INVERSE(n(4th EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b3 + c3 - a2b - a2c + abc)

X(3869) lies on these lines: {1, 21}, {2, 65}, {3, 3417}, {4, 8}, {6, 3727}, {9, 1405}, {10, 908}, {19, 2287}, {22, 3556}, {28, 1748}, {40, 78}, {41, 3496}, {46, 404}, {48, 1761}, {56, 3218}, {90, 1320}, {101, 1759}, {144, 145}, {146, 2778}, {210, 3617}, {213, 3735}, {219, 608}, {221, 394}, {314, 2995}, {354, 3622}, {391, 2262}, {392, 942}, {672, 3061}, {693, 1938}, {912, 944}, {934, 2365}, {936, 2093}, {956, 1482}, {958, 2099}, {982, 1201}, {986, 1193}, {995, 3670}, {1005, 1697}, {1156, 3680}, {1259, 3428}, {1610, 1812}, {1630, 2327}, {1698, 3754}, {1776, 2098}, {1836, 2475}, {1854, 3100}, {1898, 3621}, {2176, 3721}, {2390, 2979}, {2771, 3648}, {2802, 3632}, {3241, 3555}, {3306, 3339}, {3678, 3679}, {3698, 3740}

X(3869) = isogonal conjugate of X(2217)
X(3869) = isotomic conjugate of X(2995)
X(3869) = anticomplement of X(65)
X(3869) = X(80)-of-inner-Garcia triangle
X(3869) = radical center of circles centered at excenters and passing through corresponding vertex of anticomplementary triangle

### X(3870) = X(2886)com[INVERSE(n(3rd EULER TRIANGLE))]

Barycentrics    a(a2 + b2 + c2 - 2ab - 2ac) : :

X(3870) lies on these lines: {1, 2}, {3, 3555}, {6, 3692}, {7, 3174}, {9, 1174}, {31, 1331}, {33, 92}, {38, 2177}, {55, 63}, {57, 100}, {65, 224}, {72, 3295}, {149, 1699}, {165, 3218}, {193, 2293}, {210, 1001}, {226, 2900}, {329, 390}, {354, 1376}, {388, 3189}, {404, 3333}, {495, 3419}, {497, 908}, {528, 1836}, {664, 1088}, {902, 1707}, {950, 3436}, {960, 3303}, {962, 1490}, {968, 984}, {1005, 1697}, {1320, 3577}, {1445, 1617}, {1708, 2078}, {1709, 2801}, {2136, 3340}, {2550, 3475}, {2975, 3601}, {3242, 3666}, {3421, 3488}, {3711, 3740}

X(3870) = anticomplement of X(4847)
X(3870) = X(11550)-of-excentral-triangle
X(3870) = perspector of Gemini triangle 29 and tangential triangle, wrt ABC, of {ABC, Gemini 29}-circumconic

### X(3871) = X(12)com[INVERSE(n(4th EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(a2 + ab + ac - bc)

X(3871) lies on these lines: {1, 88}, {2, 496}, {3, 145}, {5, 149}, {8, 21}, {10, 1621}, {12, 528}, {35, 519}, {36, 3244}, {41, 644}, {56, 3241}, {60, 643}, {72, 1005}, {78, 1697}, {238, 3214}, {380, 3692}, {390, 2478}, {405, 3617}, {411, 517}, {452, 1260}, {474, 3622}, {495, 2475}, {595, 3293}, {943, 3419}, {956, 3621}, {960, 3689}, {993, 3632}, {999, 3623}, {1036, 1261}, {1329, 3058}, {1334, 3684}, {1376, 3303}, {1468, 3550}, {1470, 1476}, {2136, 3601}, {2268, 3169}, {2280, 3501}, {2346, 2550}, {2476, 3085}, {3218, 3555}, {3685, 3701}

### X(3872) = X(2886)com[INVERSE(n(4th EULER TRIANGLE))]

Trilinears    2 + csc2(A/2) : :     (Peter Moses, 1/13/2012)
Barycentrics    a(b + c - a)(b2 + c2 - a2 - 4bc) : :

X(3872) lies on these lines: {1, 2}, {9, 644}, {21, 1697}, {34, 318}, {40, 2975}, {63, 517}, {72, 1482}, {75, 77}, {100, 3576}, {104, 3359}, {149, 3586}, {280, 1219}, {355, 1532}, {391, 2324}, {392, 3305}, {404, 1420}, {515, 3434}, {518, 2099}, {529, 1836}, {908, 3421}, {946, 3436}, {952, 3419}, {958, 3057}, {960, 2098}, {993, 2802}, {996, 998}, {999, 3306}, {1060, 2968}, {1100, 3713}, {1120, 2191}, {1259, 3295}, {1319, 1376}, {2082, 2329}, {2093, 3218}, {2136, 3601}, {2345, 3554}, {2550, 3476}, {3338, 3754}

X(3872) = {X(1),X(8)}-harmonic conjugate of X(78)

### X(3873) = X(2)com[INVERSE(n(3rd EULER TRIANGLE))]

Barycentrics    a(b2 + c2 - ab - ac - bc)
X(3873) = X(381)com[Inverse(n(4th Euler triangle))]

X(3873) lies on these lines: {2, 210}, {6, 3726}, {7, 3434}, {8, 443}, {42, 982}, {43, 244}, {51, 2810}, {55, 3218}, {57, 100}, {65, 145}, {72, 3616}, {78, 3333}, {149, 1836}, {200, 3306}, {376, 517}, {404, 3338}, {614, 3315}, {674, 2979}, {748, 1757}, {938, 3436}, {940, 3242}, {960, 3622}, {962, 1071}, {984, 3720}, {999, 1260}, {1001, 3219}, {1150, 3757}, {1699, 2801}, {2280, 3509}, {2771, 3656}, {3057, 3623}, {3083, 3640}, {3084, 3641}, {3240, 3752}, {3550, 3722}, {3624, 3678}, {3632, 3754}

X(3873) = complement of X(4661)
X(3873) = anticomplement of X(210)
X(3873) = centroid of Gemini triangle 29
X(3873) = centroid of cross-triangle of Gemini triangles 1 and 13

### X(3874) = X(140)com[INVERSE(n(3rd EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b3 + c3 - a2b - a2c - 2abc)
X(3874) = X(546)com[Inverse(n(4th Euler triangle))]

X(3874) lies on these lines: {1, 21}, {2, 3678}, {4, 2801}, {7, 2894}, {8, 2891}, {10, 141}, {35, 3218}, {40, 3243}, {42, 3670}, {56, 214}, {65, 519}, {72, 354}, {78, 3338}, {92, 1844}, {100, 3336}, {145, 2802}, {210, 3634}, {213, 3726}, {244, 3216}, {295, 2809}, {386, 982}, {392, 3636}, {404, 3337}, {516, 1071}, {517, 550}, {551, 960}, {726, 2901}, {912, 946}, {997, 3333}, {1482, 2800}, {1698, 3681}, {1739, 3214}, {1759, 2280}, {3057, 3635}, {3626, 3753}

X(3874) = complement of X(5904)
X(3874) = X(13419)-of-excentral-triangle

### X(3875) = X(141)com[INVERSE(n(3rd EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2 + ab + ac - 2bc

X(3875) lies on these lines: {1, 75}, {2, 2321}, {6, 536}, {7, 145}, {8, 3672}, {9, 192}, {43, 350}, {57, 1999}, {63, 3187}, {69, 519}, {190, 1743}, {193, 527}, {269, 664}, {273, 1897}, {312, 2999}, {313, 3293}, {319, 3632}, {320, 3633}, {322, 3673}, {344, 3008}, {545, 3629}, {596, 969}, {614, 3263}, {712, 1572}, {726, 3751}, {870, 2663}, {894, 1278}, {1269, 3761}, {1423, 3169}, {1447, 3158}, {2136, 3212}, {2258, 3112}, {3216, 3264}, {3244, 3664}

### X(3876) = X(1698)com[INVERSE(n(4th EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(b2 + c2 + ab + ac + bc)

X(3876) lies on these lines: {1, 748}, {2, 72}, {3, 3219}, {8, 210}, {9, 21}, {10, 908}, {38, 978}, {43, 2292}, {63, 404}, {65, 3740}, {81, 975}, {145, 392}, {238, 976}, {329, 377}, {474, 3218}, {517, 3091}, {518, 3616}, {631, 912}, {651, 1038}, {750, 1046}, {758, 1698}, {857, 3661}, {899, 986}, {958, 3715}, {970, 3690}, {971, 3522}, {984, 1193}, {997, 2975}, {1071, 3523}, {1468, 1757}, {3061, 3691}, {3555, 3622}, {3687, 3710}

### X(3877) = X(2)com[INVERSE(n(4th EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(b2 + c2 + ab + ac - bc)
X(3877) = X(381)com[Inverse(n(3rd Euler triangle))]

X(3877) lies on these lines: {1, 21}, {2, 392}, {8, 210}, {9, 644}, {40, 404}, {65, 3616}, {72, 145}, {78, 1697}, {100, 997}, {149, 3419}, {377, 962}, {405, 1482}, {518, 1992}, {519, 3681}, {942, 3622}, {946, 2476}, {956, 3219}, {958, 2098}, {982, 1149}, {986, 1201}, {999, 3218}, {1000, 3421}, {1001, 2099}, {1334, 3061}, {2093, 3306}, {2176, 3727}, {2771, 3655}, {2800, 3576}, {2802, 3679}, {3230, 3735}, {3555, 3623}, {3624, 3754}, {3632, 3678}

### X(3878) = X(140)com[INVERSE(n(4th EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b3 + c3 - a2b - a2c + 2abc)
X(3878) = X(546)com[Inverse(n(3rd Euler triangle))]

X(3878) lies on these lines: {1, 21}, {2, 3754}, {3, 214}, {5, 10}, {8, 80}, {9, 1389}, {40, 997}, {65, 392}, {72, 519}, {101, 3496}, {210, 3626}, {213, 3727}, {354, 3636}, {404, 484}, {405, 2099}, {518, 3244}, {551, 942}, {759, 1098}, {944, 2801}, {956, 2098}, {958, 1482}, {986, 995}, {1201, 3670}, {1320, 3467}, {1829, 1904}, {2176, 3735}, {2922, 3556}, {3061, 3730}, {3230, 3721}, {3555, 3635}, {3632, 3681}, {3634, 3753}

X(3878) = complement of X(5903)
X(3878) = X(80)-of-X(1)-Brocard triangle
X(3878) = X(10)-of-inner-Garcia triangle
X(3878) = X(13403)-of-excentral-triangle
X(3878) = outer-Garcia-to-inner-Garcia similarity image of X(10)

### X(3879) = X(3739)com[INVERSE(n(3rd EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2a2 - b2 - c2 + ab + ac

X(3879) lies on these lines: {1, 69}, {2, 1449}, {6,3912} {7, 145}, {9, 193}, {10, 86}, {35, 1444}, {37, 524}, {44, 3629}, {57, 3169}, {75, 519}, {81, 306}, {100, 1014}, {141, 1100}, {142, 239}, {171, 332}, {192, 527}, {226, 1943}, {307, 1442}, {314, 1909}, {317, 1785}, {320, 3244}, {344, 1743}, {518, 3688}, {553, 3210}, {664, 3668}, {894, 2321}, {940, 3687}, {1264, 3717}, {3008, 3759}, {3241, 3672}, {3630, 3723}

X(3879) = anticomplement of X(3686)

### X(3880) = X(519)com[INVERSE(n(3rd EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(b2 + c2 + ab + ac - 4bc)
X(3880) = X(519)com[Inverse(n(4th Euler triangle))]

X(3880) lies on these lines: {1, 474}, {8, 210}, {10, 496}, {30, 511}, {36, 2932}, {37, 3169}, {65, 145}, {72, 3586}, {78, 2098}, {100, 1319}, {354, 3241}, {392, 3679}, {644, 2348}, {942, 3244}, {958, 1697}, {1145, 1737}, {1212, 3208}, {1318, 1320}, {1616, 1722}, {1875, 1897}, {1898, 3621}, {2170, 3693}, {3555, 3633}, {3616, 3698}, {3635, 3754}

X(3880) = isogonal conjugate of X(8686)
X(3880) = crossdifference of every pair of points on line X(6)X(4394)

### X(3881) = X(3628)com[INVERSE(n(3rd EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b3 + c3 - a2b - a2c - 4abc)

X(3881) lies on these lines: {1, 21}, {10, 354}, {65, 1317}, {72, 551}, {79, 149}, {100, 3337}, {244, 3293}, {517, 548}, {518, 1125}, {519, 942}, {535, 950}, {537, 3159}, {596, 740}, {946, 2801}, {960, 3636}, {2550, 3296}, {3218, 3746}, {3243, 3333}, {3624, 3681}, {3625, 3753}, {3634, 3742}

### X(3882) = X(2486)com[INVERSE(n(3rd EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a - b)(a - c)(b2 + c2 + ab + ac)

X(3882) lies on these lines: {1, 3122}, {9, 1654}, {40, 1330}, {63, 2895}, {69, 573}, {100, 109}, {101, 1310}, {163, 662}, {190, 646}, {193, 579}, {524, 2245}, {583, 3629}, {645, 3570}, {664, 1020}, {740, 1756}, {1423, 3169}, {1655, 3208}, {1763, 3719}, {1792, 3430}, {2897, 3692}

### X(3883) = X(3739)com[INVERSE(n(4th EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(2a2 + b2 + c2 + ab + ac)

X(3883) lies on these lines: {1, 69}, {8, 9}, {10, 82}, {55, 3687}, {75, 516}, {141, 1279}, {226, 3757}, {239, 3755}, {306, 1621}, {333, 643}, {519, 751}, {528, 3696}, {960, 3688}, {1001, 3416}, {1211, 3744}, {1891, 2354}, {2975, 3220}, {3058, 3706}, {3683, 3703}

### X(3884) = X(3628)com[INVERSE(n(4th EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b3 + c3 - a2b - a2c + 4abc)

X(3884) lies on these lines: {1, 21}, {10, 11}, {35, 214}, {65, 551}, {72, 3244}, {140, 517}, {210, 3625}, {405, 2098}, {518, 3635}, {519, 960}, {942, 3636}, {997, 1697}, {1000, 2551}, {1001, 1482}, {1149, 3670}, {1385, 2800}, {2170, 3294}, {3230, 3727}, {3633, 3681}

X(3884) = X(10)-of-X(1)-Brocard triangle

### X(3885) = X(1)com[INVERSE(n(4th EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(b2 + c2 + ab + ac - 3bc)
X(3885) = X(355)com[Inverse(n(3rd Euler triangle))]

X(3885) lies on these lines: {1, 88}, {8, 210}, {20, 145}, {21, 1697}, {65, 3241}, {72, 3621}, {78, 2136}, {149, 355}, {392, 3617}, {496, 1145}, {518, 3644}, {644, 2082}, {758, 3633}, {942, 3623}, {1482, 3149}, {2170, 3208}, {3622, 3753}, {3632, 3681}

### X(3886) = X(141)com[INVERSE(n(4th EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(a2 - ab - ac - 2bc)

X(3885) lies on these lines: {1, 75}, {2, 3755}, {8, 9}, {10, 344}, {55, 3706}, {69, 516}, {78, 1229}, {145, 894}, {200, 312}, {306, 3434}, {497, 3687}, {518, 3729}, {519, 1992}, {528, 3416}, {536, 3242}, {1001, 3696}, {1757, 3632}, {3210, 3677}

### X(3887) = X(2826)com[INVERSE(n(3rd EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b - c)(a2 + b2 + c2 - 2ab - 2ac + bc)
X(3887) = X(355)com[Inverse(n(4th Euler triangle))]

X(3887) lies on these lines: {1, 2254}, {10, 3716}, {11, 116}, {30, 511}, {80, 885}, {100, 101}, {103, 104}, {118, 119}, {149, 150}, {152, 153}, {214, 3126}, {663, 1734}, {1022, 1280}, {1317, 1362}, {3032, 3033}, {3036, 3041}, {3045, 3046}

X(3887) = isogonal conjugate of X(1308)
X(3887) = X(514)-of-inner-Garcia triangle

### X(3888) = X(1086)com[INVERSE(n(3rd EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a - b)(a - c)(b2 + c2 - bc)

X(3888) lies on these lines: {1, 3123}, {2, 3271}, {8, 2810}, {69, 2876}, {99, 815}, {100, 109}, {110, 833}, {190, 513}, {295, 1281}, {320, 674}, {662, 1492}, {789, 805}, {883, 926}, {1227, 2877}, {1332, 1633}, {2227, 3510}, {3056, 3662}

### X(3889) = X(1656)com[INVERSE(n(3rd EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b3 + c3 - a2b - a2c - 5abc)
X(3889) = X(3091)com[Inverse(n(4th Euler triangle))]

X(3889) lies on these lines: {1, 21}, {2, 3555}, {8, 354}, {65, 3241}, {72, 3622}, {78, 3243}, {100, 3338}, {145, 942}, {404, 3333}, {517, 3522}, {518, 3616}, {1125, 3681}, {1320, 3340}, {3218, 3295}, {3621, 3753}, {3633, 3754}

### X(3890) = X(3091)com[INVERSE(n(3rd EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b3 + c3 - a2b - a2c + 5abc)

X(3890) lies on these lines: {1, 21}, {2, 3057}, {8, 392}, {65, 3622}, {72, 3241}, {100, 1697}, {145, 960}, {210, 3621}, {517, 631}, {518, 3623}, {986, 1149}, {1001, 2098}, {1698, 2802}, {3218, 3304}, {3633, 3678}

### X(3891) = X(2887)com[INVERSE(n(3rd EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3 + ab2 + ac2 - b2c - bc2

X(3891) lies on these lines: {1, 321}, {2, 1390}, {31, 726}, {38, 1150}, {76, 3112}, {100, 1403}, {145, 388}, {192, 1621}, {239, 3681}, {278, 1280}, {518, 3187}, {536, 3744}, {1255, 3616}, {1279, 3175}, {1824, 3555}

### X(3892) = X(547)com[INVERSE(n(3rd EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b3 + c3 - a2b - a2c - 6abc)

X(3892) lies on these lines: {1, 21}, {10, 3742}, {65, 3635}, {72, 3636}, {145, 3754}, {210, 1125}, {214, 999}, {354, 519}, {518, 551}, {872, 995}, {942, 3244}, {997, 3243}, {2802, 3241}, {3158, 3333}, {3616, 3678}

### X(3893) = X(3626)com[INVERSE(n(4th EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(b2 + c2 + ab + ac - 6bc)

X(3893) lies on these lines: {1, 3689}, {8, 210}, {55, 2136}, {65, 519}, {72, 2802}, {145, 354}, {200, 2098}, {382, 517}, {392, 3626}, {518, 1278}, {942, 3633}, {1697, 3683}, {1706, 3304}, {3244, 3753}

### X(3894) = X(549)com[INVERSE(n(3rd EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2b3 + 2c3 - 2a2b - 2a2c - 3abc)

X(3894) lies on these lines: {1, 21}, {46, 3158}, {65, 3632}, {72, 3624}, {78, 3337}, {210, 942}, {517, 3534}, {518, 599}, {912, 1699}, {2093, 3174}, {3555, 3633}

### X(3895) = X(495)com[INVERSE(n(4th EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(a2 + b2 + c2 + 2ab + 2ac - 4bc)

X(3895) lies on these lines: {1, 88}, {8, 9}, {40, 145}, {46, 3244}, {57, 3241}, {63, 519}, {78, 3057}, {960, 3711}, {1706, 3616}, {1837, 3036}, {2320, 3601}, {3305, 3679}, {3333, 3623}, {3338, 3635}

### X(3896) = X(3741)com[INVERSE(n(3rd EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(a2 - bc)

X(3896) lies on these lines: {2, 3696}, {10, 1962}, {38, 519}, {42, 321}, {55, 3187}, {65, 145}, {100, 1402}, {192, 3681}, {239, 1621}, {306, 3755}, {312, 3240}, {386, 3702}, {2901, 3293}

### X(3897) = X(2476)com[INVERSE(n(4th EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(b2 + c2 - 2a2 - ab - ac - 3bc)

X(3897) lies on these lines: {1, 21}, {2, 355}, {8, 2320}, {100, 3612}, {214, 1698}, {388, 1319}, {404, 3576}, {452, 3487}, {515, 2476}, {958, 3715}, {1001, 1388}, {1320, 1697}, {2136, 3601}

### X(3898) = X(547)com[INVERSE(n(4th EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b3 + c3 - a2b - a2c + 6abc)

X(3898) lies on these lines: {1, 21}, {2, 2802}, {10, 496}, {55, 214}, {65, 3636}, {72, 3635}, {145, 3678}, {210, 392}, {517, 549}, {960, 3244}, {997, 3158}, {1125, 3057}, {3616, 3754}

### X(3899) = X(549)com[INVERSE(n(4th EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2b3 + 2c3 - 2a2b - 2a2c + 3abc)

X(3899) lies on these lines: {1, 21}, {8, 3583}, {65, 3624}, {72, 3586}, {165, 2800}, {210, 381}, {392, 3742}, {484, 997}, {960, 1698}, {1376, 3245}, {2802, 3681}, {3057, 3633}

### X(3900) = X(514)com[INVERSE(n(3rd EULER TRIANGLE))]

Trilinears    directed distance from A to Soddy line : :
Barycentrics   a(b - c)(b + c - a)2 : :
X(3900) = X(514)com[Inverse(n(4th Euler triangle))]

X(3900) lies on these lines: {1, 905}, {8, 885}, {30, 511}, {55, 1946}, {650, 663}, {764, 3680}, {1146, 3022}, {1960, 2516}, {2192, 2431}, {2254, 3669}, {3064, 3700}, {3158, 3251}

X(3900) = isogonal conjugate of X(934)
X(3900) = isotomic conjugate of X(4569)

### X(3901) = X(550)com[INVERSE(n(3rd EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2b3 + 2c3 -2a2b - 2a2c - abc)
X(3901) = X(3627)com[Inverse(n(4th Euler triangle))]

X(3901) lies on these lines: {1, 21}, {65, 3679}, {72, 1698}, {78, 3336}, {79, 3419}, {517, 1657}, {518, 3632}, {519, 1770}, {942, 3624}, {997, 3337}, {3681, 3754}

### X(3902) = X(3741)com[INVERSE(n(4th EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c - a)(b + c + 4a)

X(3902) lies on these lines: {8, 210}, {75, 3241}, {314, 1320}, {321, 519}, {528, 1227}, {1089, 3625}, {1441, 2099}, {2170, 2321}

### X(3903) = X(115)com[INVERSE(n(3rd EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a - b)(a - c)(b2 + ac)(c2 + ab)
X(3903) = X(114)com[Inverse(n(4th Euler triangle))]

X(3903) lies on these lines: {1, 1581}, {99, 512}, {256, 1320}, {257, 3057}, {643, 3573}, {893, 3744}, {1120, 1431}, {1280, 1432}

X(3903) = isogonal conjugate of X(4367)
X(3903) = isotomic conjugate of X(4374)
X(3903) = trilinear pole of line X(9)X(43)

### X(3904) = X(3716)com[INVERSE(n(4th EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)(b + c - a)(b2 + c2 - a2 - bc)

X(3904) lies on these lines: {190, 644}, {323, 401}, {514, 661}, {676, 3616}, {1320, 2804}, {2254, 2785}, {2401, 2990}

X(3904) = isotomic conjugate of X(665)

### X(3905) = X(626)com[INVERSE(n(3rd EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4 - ab3 - ac3 - b3c - bc3

X(3905) lies on these lines: {1, 75}, {32, 712}, {192, 2329}, {239, 3061}, {726, 1975}, {1429, 3210}, {1959, 3187}

### X(3906) = X(523)com[INVERSE(n(5th EULER TRIANGLE))]

Barycentrics   (b2 - c2)(2b2 + 2c2 - a2) : :

X(3906) lies on these lines: {30, 511}, {39, 647}, {76, 850}, {262, 2394}, {879, 3431}

X(3906) = infinite point of radical axis of Brocard circle and orthocentroidal circle
X(3906) = crossdifference of every pair of points on line X(6)X(23)

### X(3907) = X(512)com[INVERSE(n(3rd EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)(b + c - a)(a2 + bc)
X(3907) = X(512)com[Inverse(n(4th Euler triangle))]

X(3907) lies on these lines: {1, 810}, {30, 511}, {663, 3716}, {1027, 1222}, {1459, 2517}

X(3907) = crossdifference of every pair of points on line X(6)X(893)

### X(3908) = X(120)com[INVERSE(n(5th EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a - b)(a - c)(2b2 + 2c2 - a2)

X(3908) lies on these lines: {100, 110}, {513, 644}, {668, 892}

### X(3909) = X(3120)com[INVERSE(n(3rd EULER TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a - b)(a - c)(b3 + c3 + ab2 + ac2)

X(3909) lies on these lines: {100, 109}, {2836, 2895}

### X(3910) = X(830)com[INVERSE(n(3rd EULER TRIANGLE))]

Barycentrics   (b - c)(b + c - a)(b2 + c2 + ab + ac) : :

X(3910) lies on this line: {30, 511}

X(3910) = isogonal conjugate of X(8687)
X(3910) = isotomic conjugate of X(6648)
X(3910) = crossdifference of every pair of points on line X(2)X(12)
X(3910) = X(830)com[Inverse(n(4th Euler triangle))]

### X(3911) = X(3689)com[INTOUCH TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (2a - b - c)(a + b - c)(a - b + c)
X(3911) = 9r*X(2) - (r + 4R)*X(7)

X(3911) lies on these lines: {1, 631}, {2, 7}, {3, 950}, {8, 1420}, {10, 56}, {11, 516}, {12, 3634}, {36, 80}, {40, 3086}, {46, 499}, {65, 392}, {73, 3216}, {88, 655}, {100, 2078}, {108, 1861}, {109, 238}, {140, 942}, {165, 497}, {201, 3670}, {208, 475}, {214, 519}, {216, 1108}, {241, 514}, {244, 3011}, {319, 2985}, {333, 1412}, {376, 3586}, {388, 1698}, {405, 1466}, {468, 1876}, {484, 3582}, {496, 3579}, {498, 3338}, {517, 1387}, {518, 3035}, {551, 2099}, {580, 3075}, {602, 1771}, {603, 1724}, {604, 1150}, {673, 927}, {750, 1471}, {899, 1458}, {902, 1647}, {914, 1813}, {936, 1467}, {938, 3523}, {993, 1470}, {1016, 1429}, {1054, 1738}, {1279, 3756}, {1357, 1463}, {1371, 1659}, {1376, 1617}, {1388, 3244}, {1402, 3741}, {1436, 1751}, {1519, 2950}, {1621, 3256}, {1699, 3474}, {1768, 1776}, {1838, 1940}, {3085, 3333}, {3321, 3323}, {3339, 3485}, {3340, 3616}, {3476, 3679}, {3487, 3525}, {3488, 3524}

X(3911) = isogonal conjugate of X(2316)
X(3911) = isotomic conjugate of X(4997)
X(3911) = complement of X(908)
X(3911) = {X(2),X(7)}-harmonic conjugate of X(5219)
X(3911) = {X(2),X(57)}-harmonic conjugate of X(226)
X(3911) = inverse-in-circumconic-centered-at-X(9) of X(57)
X(3911) = perspector of Gemini triangle 10 and cross-triangle of ABC and Gemini triangle 10
X(3911) = trilinear pole of line X(900)X(1317) (the perspectrix of ABC and Gemini triangle 9)

### X(3912) = X(3747)com[INTOUCH TRIANGLE))]

Barycentrics    b2 + c2 - ab - ac : :

X(3912) is the perspector of the circumconic passing through the isogonal conjugates of PU(48). (Randy Hutson, August 29, 2018)

X(3912) lies on these lines: {1, 2}, {7, 346}, {6,3879}, {9, 69}, {37, 141}, {44, 524}, {45, 599}, {57, 345}, {63, 3730}, {75, 142}, {76, 85}, {100, 2725}, {101, 2862}, {144, 3161}, {190, 320}, {192, 3662}, {193, 1743}, {241, 3693}, {297, 1785}, {319, 3686}, {321, 1930}, {333, 1174}, {334, 350}, {335, 726}, {354, 3703}, {514, 661}, {516, 3685}, {518, 3717}, {536, 1086}, {594, 3739}, {666, 2338}, {740, 1738}, {894, 3664}, {942, 3695}, {980, 2276}, {1001, 3416}, {1016, 1429}, {1100, 3589}, {1155, 3712}, {1229, 1441}, {1332, 2323}, {1438, 3684}, {1445, 3692}, {1449, 3618}, {1500, 3666}, {1708, 3719}, {1818, 1861}, {2228, 3122}, {2239, 3747}, {3061, 3452}, {3175, 3782}, {3247, 3619}, {3620, 3731}

X(3912) = isogonal conjugate of X(1438)
X(3912) = isotomic conjugate of X(673)
X(3912) = complement of X(239)
X(3912) = inverse-in-Steiner-circumellipse of X(8)
X(3912) = inverse-in-Steiner-inellipse of X(10)
X(3912) = X(6)-isoconjugate of X(105)
X(3912) = trilinear pole of line X(918)X(2254)
X(3912) = crossdifference of PU(48)
X(3912) = crossdifference of every pair of points on line X(31)X(649)
X(3912) = X(2)-Ceva conjugate of X(17755)
X(3912) = barycentric square root of X(4437)

### X(3913) = X(10)com[INVERSE(n(INTOUCH TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(a2 + ab + ac - 2bc)

Let OA be the circle tangent to side BC at its midpoint and to the circumcircle on the same side of BC as A. Define OB and OC cyclically. X(3913) is the radical center of the circles OA, OB, OC. See the reference at X(1001).

X(3913) lies on these lines: {1, 474}, {2, 3303}, {3, 519}, {4, 528}, {6, 979}, {8, 21}, {10, 1001}, {12, 3434}, {20, 529}, {35, 956}, {36, 3633}, {40, 518}, {43, 1191}, {46, 3555}, {56, 100}, {65, 224}, {72, 2900}, {78, 3057}, {81, 2334}, {200, 960}, {218, 1018}, {220, 3208}, {341, 3685}, {390, 480}, {404, 3241}, {405, 3679}, {497, 1329}, {521, 3157}, {535, 1657}, {950, 1260}, {978, 1616}, {986, 3242}, {993, 3625}, {999, 3244}, {1104, 3749}, {1279, 1722}, {1320, 1392}, {1466, 3476}, {1482, 2802}, {1621, 3617}, {2082, 3693}, {2264, 3692}, {2269, 3713}, {2478, 3058}, {2550, 2894}, {2886, 3085}, {2975, 3621}, {3035, 3086}, {3243, 3339}, {3698, 3748}

X(3913) = X(6247)-of-excentral-triangle

### X(3914) = X(31)com[ORTHIC TRIANGLE))]

Barycentrics    (b + c)(a2 + b2 + c2 - 2bc) : :

X(3914) lies on these lines: {1, 224}, {2, 968}, {4, 1039}, {6, 1836}, {10, 321}, {11, 3752}, {31, 516}, {38, 3663}, {42, 226}, {43, 908}, {46, 1076}, {55, 3011}, {58, 1770}, {63, 1711}, {65, 225}, {75, 305}, {141, 3706}, {142, 3720}, {228, 1284}, {278, 2263}, {306, 740}, {307, 3778}, {354, 1086}, {387, 1838}, {497, 614}, {517, 1072}, {518, 3782}, {528, 3744}, {536, 3703}, {612, 2550}, {748, 3008}, {774, 1210}, {851, 1402}, {899, 3452}, {942, 1070}, {946, 1193}, {1211, 3696}, {1279, 3058}, {1699, 2999}, {1722, 2478}, {1837, 1853}, {1842, 3556}, {1851, 2082}, {2006, 3256}, {2092, 3136}, {2650, 3671}, {2886, 3666}, {3210, 3705}

### X(3915) = X(3710)com[TANGENTIAL TRIANGLE))]

Trilinears        a(as - bc) : b(bs - ca ) : c(cs - ab)
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a2 + ab + ac - 2bc)

X(3915) lies on these lines: {1, 21}, {3, 902}, {6, 1334}, {8, 238}, {10, 748}, {32, 3230}, {35, 995}, {40, 614}, {41, 1914}, {42, 3295}, {46, 244}, {48, 3285}, {55, 1191}, {56, 1149}, {65, 1279}, {78, 3749}, {100, 978}, {109, 1106}, {171, 3616}, {212, 1104}, {213, 2241}, {221, 1458}, {386, 2177}, {392, 1472}, {404, 3550}, {517, 582}, {519, 1724}, {601, 1385}, {603, 1319}, {750, 1125}, {944, 3073}, {946, 3011}, {960, 976}, {1001, 1918}, {1042, 1617}, {1055, 3053}, {1253, 1697}, {1388, 1399}, {1451, 2099}, {1829, 2212}, {1834, 3058}, {1935, 3476}, {2098, 2361}, {2268, 2300}

### X(3916) = X(35)com[INVERSE(n(TANGENTIAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2a + b + c)(a2 - b2 - c2)

X(3916) lies on these lines: {1, 3052}, {3, 63}, {8, 376}, {9, 474}, {10, 535}, {20, 3419}, {21, 942}, {35, 518}, {36, 191}, {40, 956}, {44, 3216}, {46, 958}, {55, 3555}, {56, 392}, {57, 405}, {58, 3666}, {65, 993}, {140, 908}, {144, 3523}, {283, 1789}, {329, 631}, {404, 3219}, {411, 971}, {517, 2975}, {553, 1125}, {603, 1214}, {758, 2646}, {896, 1193}, {988, 1707}, {1001, 3338}, {1104, 3670}, {1158, 3428}, {1212, 1759}, {1376, 3697}, {1437, 1444}, {1465, 1935}, {1466, 1708}, {1724, 3752}, {1748, 1871}, {1770, 2886}, {2094, 3616}, {2915, 3220}, {3337, 3742}

### X(3917) = X(2)com[INVERSE(n(TANGENTIAL TRIANGLE))]

Barycentrics    a2(b2 + c2)(b2 + c2 - a2) : :
Trilinears    (cos A) (b^2 + c^2) : :
Trilinears    2 b c - a b cos B - a c cos C : :
Trilinears    cot A sin(A + ω) : :

X(3917) lies on these lines: {2, 51}, {3, 49}, {22, 1495}, {25, 1350}, {38, 1401}, {39, 3051}, {52, 140}, {63, 295}, {69, 305}, {125, 343}, {141, 427}, {143, 632}, {165, 2807}, {181, 750}, {182, 1993}, {187, 1501}, {212, 1364}, {216, 3289}, {219, 1473}, {228, 1818}, {354, 674}, {389, 631}, {404, 970}, {418, 2972}, {426, 577}, {549, 1154}, {575, 1994}, {599, 1853}, {612, 1469}, {614, 3056}, {626, 2450}, {748, 3271}, {851, 1764}, {991, 1011}, {1038, 1425}, {1040, 3270}, {1194, 1613}, {1196, 3231}, {1352, 1370}, {1915, 2076}, {2810, 3681}, {2896, 3491}, {3525, 3567}

X(3917) = anticomplement of X(5943)
X(3917) = {X(2),X(51)}-harmonic conjugate of X(373)
X(3917) = crossdifference of every pair of points on line X(419)X(2501) (radical axis of circumcircle and orthosymmedial circle)
X(3917) = centroid of pedal triangle of X(20)

### X(3918) = X(140)com[INVERSE(n(HEXYL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(b2 + c2 - a2 - 5bc)

X(3918) lies on these lines: {10, 12}, {354, 3625}, {474, 1388}, {484, 3647}, {517, 3628}, {942, 3626}, {1125, 1387}, {3635, 3742}

### X(3919) = X(381)com[INVERSE(n(HEXYL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(2b2 + 2c2 - 2a2 - 5bc)

X(3919) lies on these lines: {1, 1392}, {10, 12}, {354, 2802}, {517, 549}, {942, 3244}, {1159, 1376}, {1621, 3245}

### X(3920) = X(3745)com[INVERSE(n(INTOUCH TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 + b2 + c2 + bc)

X(3920) lies on these lines: {1, 2}, {6, 3681}, {22, 55}, {23, 3743}, {25, 3295}, {31, 984}, {33, 390}, {37, 82}, {38, 171}, {81, 518}, {86, 3263}, {100, 3666}, {105, 1255}, {172, 1627}, {210, 1386}, {238, 756}, {388, 1370}, {427, 495}, {611, 1993}, {750, 982}, {846, 902}, {940, 3242}, {968, 3749}, {1038, 3600}, {1056, 1060}, {1150, 3769}, {1172, 2346}, {1180, 2276}, {1194, 1500}, {1203, 3678}, {1469, 2979}, {1757, 2308}, {1921, 3112}, {1962, 3722}, {1995, 3303}, {3056, 3060}, {3290, 3723}, {3306, 3677}, {3315, 3742}

### X(3921) = X(3524)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(b2 + c2 - a2 + 8bc)

X(3921) lies on these lines: {10, 12}, {392, 3679}, {405, 3158}, {517, 3545}, {1698, 3555}, {1962, 3214}

### X(3922) = X(3090)com[INVERSE(n(HEXYL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(3b2 + 3c2 - 3a2 - 10bc)

X(3922) lies on these lines: {10, 12}, {145, 354}, {517, 3526}, {942, 3632}, {1706, 3689}, {3057, 3616}

### X(3923) = X(1)com[1st BROCARD TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3 + b2c + bc2

X(3923) lies on these lines: {1, 87}, {2, 846}, {4, 9}, {6, 740}, {8, 1757}, {20, 2944}, {31, 321}, {44, 3696}, {55, 1215}, {58, 314}, {63, 3741}, {75, 238}, {76, 1966}, {171, 312}, {182, 2783}, {190, 984}, {226, 3771}, {256, 1008}, {386, 1045}, {519, 1992}, {536, 1386}, {537, 3242}, {752, 3416}, {896, 1150}, {936, 1721}, {964, 2292}, {988, 1125}, {990, 997}, {996, 2802}, {1009, 1284}, {1352, 2792}, {1468, 3702}, {1836, 2887}, {2308, 3187}, {2795, 3734}, {3175, 3745}

X(3923) = X(1)-of-1st-Brocard-triangle

### X(3924) = X(3710)com[INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a3 + b3 + c3 - b2c - bc2)

X(3924) lies on these lines: {1, 2}, {4, 3120}, {6, 2294}, {7, 2647}, {21, 986}, {28, 2206}, {31, 65}, {32, 3125}, {34, 207}, {38, 958}, {40, 902}, {56, 244}, {81, 409}, {227, 1319}, {405, 2292}, {517, 582}, {604, 1880}, {607, 2170}, {748, 960}, {758, 1724}, {942, 1468}, {977, 1220}, {982, 2975}, {993, 3670}, {1106, 1455}, {1148, 1870}, {1191, 2099}, {1253, 1279}, {1453, 2308}, {1616, 2098}, {1837, 3772}, {1854, 2310}, {2331, 3554}, {2646, 3752}

### X(3925) = X(1006)com[INVERSE(n(HEXYL TRIANGLE))]

Barycentrics    (b + c)(b2 + c2 - ab - ac - 2bc) : :

Let A'B'C' be the intouch triangle of the extangents triangle, if ABC is acute. Then A'B'C' is homothetic to the medial triangle at X(3925). (Randy Hutson, December 2, 2017)

X(3925) lies on these lines: {2, 11}, {5, 40}, {8, 3475}, {9, 1836}, {10, 12}, {19, 427}, {38, 1086}, {56, 443}, {71, 1213}, {75, 3703}, {125, 3611}, {141, 3779}, {142, 354}, {306, 3696}, {329, 3715}, {377, 958}, {429, 1869}, {495, 3679}, {496, 3624}, {516, 3683}, {594, 2294}, {612, 3772}, {756, 3120}, {858, 3101}, {908, 3740}, {984, 3782}, {1212, 1855}, {1329, 2476}, {1738, 3666}, {1842, 1883}, {1853, 3197}, {1859, 1861}, {3189, 3616}

### X(3926) = X(3053)com[INVERSE(n(TANGENTIAL TRIANGLE))]

Trilinears    cot2A csc A : : (Peter Moses, 1/13/2012)
Barycentrics    cot2 A : cot2 B : cot2 C
Barycentrics    (b2 + c2 - a2)2 : :

X(3926) lies on these lines: {2, 39}, {3, 69}, {4, 325}, {5, 1007}, {20, 99}, {32, 193}, {110, 2366}, {115, 2996}, {183, 631}, {187, 439}, {264, 1217}, {276, 1502}, {304, 345}, {316, 3146}, {326, 1264}, {339, 3548}, {350, 3086}, {441, 1073}, {491, 1587}, {492, 1588}, {498, 3761}, {499, 3760}, {524, 3053}, {574, 3620}, {626, 2549}, {1078, 3523}, {1102, 3719}, {1235, 3541}, {1310, 3556}, {1909, 3085}, {2548, 3734}

X(3926) = isogonal conjugate of X(2207)
X(3926) = isotomic conjugate of X(393)
X(3926) = complement of X(6392)
X(3926) = anticomplement of X(3767)
X(3926) = X(92)-isoconjugate of X(1974)
X(3926) = trilinear pole of line X(520)X(3265)
X(3926) = trilinear product of vertices of pedal triangle of X(2) reflected in X(2)
X(3926) = pole wrt polar circle of trilinear polar of X(6524) (line X(2489)X(2508))
X(3926) = polar conjugate of X(6524)
X(3926) = barycentric square of X(69)
X(3926) = trilinear product of vertices of X(2)-anti altimedial triangle

### X(3927) = X(1)com[INVERSE(n(EXTANGENTS TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2b + 2c + a)(b2 + c2 - a2)

X(3927) lies on these lines: {1, 3683}, {3, 63}, {4, 144}, {5, 329}, {8, 30}, {9, 942}, {10, 527}, {40, 971}, {46, 210}, {55, 191}, {69, 3695}, {200, 3579}, {201, 222}, {219, 3157}, {382, 3419}, {405, 3219}, {474, 3218}, {500, 3190}, {518, 3295}, {595, 3242}, {758, 958}, {894, 2049}, {896, 976}, {908, 1656}, {956, 1482}, {960, 999}, {984, 1046}, {986, 1757}, {1376, 3678}, {1698, 3715}, {1714, 3782}

X(3927) = Conway-triangle-to-inner-Conway-triangle similarity image of X(3)

### X(3928) = X(154)com[INVERSE(n(INTANGENTS TRIANGLE))]

Trilinears       t(A,B,C) : t(B,C,A) : t(C,A,B), where t(A,B,C) = 3 cot A - csc A    (Peter Moses, 1/13/2012)
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(3a2 - 3b2 - 3c2 + 2bc)

X(3928) lies on these lines: {1, 3052}, {2, 7}, {30, 84}, {31, 3677}, {40, 376}, {46, 529}, {55, 3243}, {165, 518}, {191, 3338}, {200, 1155}, {219, 1407}, {222, 2323}, {528, 1768}, {536, 1764}, {551, 3333}, {614, 896}, {758, 3576}, {940, 3247}, {956, 2093}, {958, 3339}, {960, 3361}, {982, 1707}, {988, 1046}, {1449, 3666}, {1453, 3670}, {1697, 3241}, {1743, 3752}, {1761, 2257}, {2975, 3340}, {3359, 3654}

X(3928) = centroid of Gemini triangle 24

### X(3929) = X(154)com[INVERSE(n(EXTANGENTS TRIANGLE))]

Trilinears       t(A,B,C) : t(B,C,A) : t(C,A,B), where t(A,B,C) = 3 cot A + csc A    (Peter Moses, 1/13/2012)
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(3a2 - 3b2 - 3c2 - 2bc)

X(3929) lies on these lines: {1, 3683}, {2, 7}, {10, 3474}, {30, 40}, {44, 2999}, {72, 3601}, {81, 3247}, {84, 376}, {165, 210}, {201, 1394}, {219, 2003}, {220, 222}, {238, 3677}, {333, 3729}, {519, 1697}, {612, 896}, {728, 3719}, {846, 3751}, {912, 3576}, {940, 3731}, {958, 3340}, {960, 1420}, {984, 1707}, {991, 2318}, {1155, 3715}, {1214, 1419}, {1621, 3243}, {1743, 3666}, {3158, 3681}, {3534, 3587}

X(3929) = 2nd-extouch-to-excentral similarity image of X(2)

### X(3930) = X(672)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(b2 + c2 - ab - ac)

X(3930) lies on these lines: {1, 1390}, {9, 1174}, {37, 42}, {38, 2276}, {72, 1334}, {100, 3509}, {101, 2752}, {145, 3061}, {226, 306}, {244, 1575}, {518, 672}, {519, 2170}, {523, 661}, {594, 2294}, {678, 2243}, {758, 1018}, {899, 3290}, {910, 3689}, {1400, 3694}, {1475, 3555}, {1500, 2292}, {1914, 3722}, {1926, 1978}, {2246, 3684}, {2295, 2650}, {2340, 2356}, {2345, 3475}, {3294, 3678}

### X(3931) = X(581)com[INVERSE(n(HEXYL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(a2 + b2 + c2 + 2ab + 2ac)

X(3931) lies on these lines: {1, 3}, {2, 3702}, {4, 941}, {10, 37}, {38, 3555}, {42, 72}, {210, 3293}, {226, 227}, {386, 960}, {392, 1193}, {405, 968}, {595, 1386}, {756, 3214}, {975, 1376}, {976, 2177}, {984, 1716}, {1089, 3175}, {1100, 2241}, {1125, 3752}, {1181, 1409}, {1427, 3671}, {1441, 3672}, {1465, 3485}, {1706, 3247}, {1724, 3683}, {1785, 1882}, {1962, 3753}, {2887, 3178}

### X(3932) = X(1279)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b2 + c2 - ab - ac)

X(3932) lies on these lines: {1, 3589}, {2, 1390}, {8, 344}, {9, 3416}, {10, 37}, {11, 3006}, {12, 313}, {23, 100}, {65, 3710}, {120, 3263}, {141, 984}, {210, 306}, {312, 2886}, {345, 1376}, {346, 2550}, {442, 1089}, {516, 2325}, {518, 3717}, {519, 1279}, {523, 1577}, {524, 1757}, {528, 3685}, {536, 1738}, {726, 1086}, {756, 1211}, {1861, 3693}, {3687, 3740}

### X(3933) = X(32)com[INVERSE(n(TANGENTIAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 + c2)(b2 + c2 - a2)
Barycentrics    csc A cot A sin(A + ω) : :

X(3933) lies on these lines: {3, 69}, {5, 76}, {11, 3760}, {12, 3761}, {30, 315}, {32, 524}, {39, 141}, {99, 550}, {140, 183}, {187, 3630}, {194, 3314}, {264, 1595}, {304, 337}, {305, 1368}, {316, 3627}, {350, 496}, {394, 441}, {427, 1235}, {495, 1909}, {538, 626}, {549, 1078}, {574, 3631}, {980, 1211}, {1007, 1656}, {1353, 3398}, {1930, 3665}, {3673, 3705}

X(3933) = crossdifference of every pair of points on line X(2489)X(3804)

### X(3934) = X(2023)com(1st BROCARD TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2b2c2 + c2a2 + a2b2
X(3934) = 3*X(2) + X(76)

X(3934) lies on these lines: {2, 39}, {3, 3734}, {5, 141}, {32, 183}, {69, 2548}, {75, 1574}, {83, 385}, {140, 620}, {187, 384}, {230, 736}, {232, 1235}, {262, 3090}, {264, 3199}, {316, 2896}, {325, 1506}, {350, 1500}, {574, 1975}, {726, 3634}, {730, 1125}, {732, 3589}, {1015, 1909}, {1207, 3499}, {1656, 3095}, {2275, 3761}, {2276, 3760}, {3094, 3763}

X(3934) = centroid of ABCX(76)
X(3934) = Kosnita(X(76),X(2))
X(3934) = complement of X(39)
X(3934) = perspector of the medial triangle and the tangential triangle, wrt the medial triangle, of the bicevian conic of X(2) and X(6)

### X(3935) = X(1155)com[INVERSE(n(INTOUCH TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 + b2 + c2 - 2ab - 2ac + bc)

X(3935) lies on these lines: {1, 2}, {44, 765}, {55, 1776}, {63, 3158}, {72, 1005}, {88, 1280}, {100, 518}, {144, 3174}, {149, 908}, {210, 1621}, {238, 3722}, {329, 2900}, {404, 3555}, {522,4724}, {678, 896}, {740, 1109}, {756, 3750}, {758, 3245}, {902, 1757}, {984, 2177}, {1001, 3711}, {1071, 3579}, {2246, 3684}, {3189, 3436}, {3243, 3306}, {3678, 3746}, {3740, 3748}

X(3935) = inverse-in-circumconic-centered-at-X(1) of X(2)
X(3935) = endo-homothetic center of X(2)- and X(3)-Ehrmann triangles; the homothetic center is X(15139)

### X(3936) = X(902)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b2 + c2 - a2 - bc)

X(3936) lies on these lines: {1, 3454}, {2, 6}, {8, 442}, {10, 2650}, {21, 1330}, {31, 3771}, {42, 2887}, {100, 851}, {145, 1834}, {226, 306}, {304, 1228}, {312, 1230}, {320, 2245}, {329, 440}, {346, 1901}, {514, 661}, {518, 3006}, {740, 3120}, {752, 902}, {856, 1809}, {860, 1870}, {1043, 2475}, {2092, 3662}, {2292, 3178}, {3187, 3772}, {3649, 3704}

X(3936) = isotomic conjugate of X(24624)
X(3936) = trilinear pole of line X(4707)X(4736) (the tangent at X(4736) to the inellipse centered at X(10))

### X(3937) = X(11)com[INVERSE(n(INTANGENTS TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b - c)2(b2 + c2 - a2)

X(3937) lies on these lines: {1, 2841}, {3, 1331}, {11, 513}, {25, 1407}, {31, 1401}, {51, 57}, {63, 295}, {100, 2810}, {104, 2818}, {123, 125}, {184, 222}, {214, 2842}, {244, 1357}, {373, 3306}, {511, 3218}, {512, 2611}, {603, 1425}, {1015, 1977}, {1086, 2969}, {1260, 1810}, {1319, 2390}, {1364, 3270}, {1398, 1413}, {1463, 3011}, {1495, 3220}, {1768, 2807}

### X(3938) = X(3744)com[INVERSE(n(INTOUCH TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 + b2 + c2 - ab - ac)

X(3938) lies on these lines: {1, 2}, {31, 518}, {33, 1851}, {37, 2280}, {38, 55}, {63, 902}, {100, 982}, {210, 748}, {238, 3681}, {244, 1376}, {354, 750}, {528, 3782}, {756, 1001}, {896, 3052}, {984, 1621}, {1468, 3555}, {1807, 3478}, {2177, 3666}, {2292, 3295}, {2308, 3751}, {3120, 3434}, {3158, 3677}, {3218, 3550}, {3689, 3752}

### X(3939) = X(1086)com[INVERSE(n(INTOUCH TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a - b)(a - c)(b + c - a)

X(3939) lies on these lines: {3, 2810}, {9, 294}, {31, 678}, {40, 2835}, {55, 2316}, {59, 677}, {100, 109}, {101, 692}, {190, 522}, {200, 212}, {210, 2328}, {219, 480}, {284, 2311}, {573, 2876}, {643, 645}, {1018, 1783}, {1026, 1332}, {1461, 2283}, {1471, 3243}, {1724, 3189}, {1743, 3174}, {2209, 3169}, {2323, 2340}, {2361, 3689}

X(3939) = isogonal conjugate of X(3676)
X(3939) = trilinear pole of line X(41)X(55)
X(3939) = crossdifference of every pair of points on line X(1086)X(1358)
X(3939) = perspector of anticevian triangle of X(101) and unary cofactor triangle of intangents triangle

### X(3940) = X(999)com[INVERSE(n(TANGENTIAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2b + 2c - a)(b2 + c2 - a2)

X(3940) lies on these lines: {1, 210}, {3, 63}, {5, 8}, {30, 329}, {69, 1565}, {144, 376}, {200, 517}, {219, 1807}, {381, 908}, {480, 3428}, {518, 997}, {519, 3452}, {758, 1376}, {936, 942}, {952, 3421}, {956, 3681}, {958, 3678}, {960, 3295}, {995, 3242}, {1064, 2340}, {1159, 3753}, {1265, 3695}, {2098, 3632}, {2099, 3679}

### X(3941) = X(3694)com(TANGENTIAL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(b2 + c2 - bc - ca - ab)

X(3941) lies on these lines: {1, 3286}, {3, 1386}, {6, 2223}, {31, 48}, {36, 1631}, {55, 1100}, {56, 1279}, {100, 3759}, {513, 1423}, {579, 674}, {583, 3779}, {604, 692}, {665, 2498}, {934, 2369}, {1009, 3416}, {1011, 3745}, {1402, 3052}, {1444, 1621}, {1445, 2283}, {1617, 3433}, {2209, 3248}, {2245, 3056}, {2260, 2293}

### X(3942) = X(2310)com[INVERSE(n(INTANGENTS TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b - c)2(b2 + c2 - a2)

X(3942) lies on these lines: {1, 1633}, {7, 1953}, {19, 269}, {38, 2877}, {48, 77}, {56, 2097}, {57, 909}, {63, 1332}, {69, 337}, {241, 2183}, {244, 659}, {320, 1959}, {513, 2310}, {651, 2265}, {1086, 1358}, {1108, 1122}, {1364, 3270}, {1418, 2262}, {1419, 2261}, {1422, 1435}, {1443, 2173}, {2294, 3664}, {3271, 3675}

### X(3943) = X(44)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b + c - 2a)

X(3943) lies on these lines: {6, 145}, {8, 45}, {9, 3632}, {10, 37}, {44, 519}, {101, 2758}, {141, 192}, {190, 524}, {306, 3175}, {320, 545}, {523, 661}, {536, 1086}, {1018, 2245}, {1100, 3635}, {1317, 1404}, {1897, 1990}, {2171, 3649}, {2345, 3616}, {3247, 3624}, {3625, 3707}, {3636, 3723}, {3644, 3662}, {3672, 3763}

### X(3944) = X(3550)com(INTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b3 + c3 - b2c - bc2 + abc

X(3944) lies on these lines: {1, 4}, {2, 846}, {5, 986}, {11, 982}, {43, 908}, {79, 987}, {115, 3735}, {171, 1836}, {238, 3772}, {312, 2887}, {329, 1757}, {516, 3550}, {726, 3705}, {752, 3769}, {984, 2886}, {1044, 1076}, {1738, 3452}, {1756, 1764}, {2292, 2476}, {3496, 3767}, {3551, 3662}, {3685, 3771}

### X(3945) = X(3247)com(INTOUCH TRIANGLE)

Barycentrics   b2 + c2 - 3a2 - 2ab - 2ac - 2bc : :

X(3945) lies on these lines: {1, 7}, {2, 6}, {3, 1014}, {37, 144}, {57, 2269}, {75, 145}, {142, 1449}, {171, 1253}, {226, 1419}, {319, 3617}, {320, 1279}, {344, 3758}, {346, 894}, {354, 3056}, {522, 4724}, {527, 3247}, {941, 980}, {1002, 3779}, {1418, 3666}, {1433, 1440}, {1434, 3522}, {1456, 3485}, {3475, 3745}

X(3945) = intersection of tangents at X(2) and X(7) to Lucas cubic K007
X(3935) = inverse-in-circumconic-centered-at-X(1) of X(2)
X(3935) = endo-homothetic center of X(2)- and X(3)-Ehrmann triangles; the homothetic center is X(15139)

### X(3946) = X(1386)com[INVERSE(n(HEXYL TRIANGLE))]

Barycentrics    2a2 + b2 + c2 + ab + ac - 2bc : :

X(3946) lies on these lines: {1, 142}, {2, 2321}, {6, 527}, {7, 1419}, {9, 3672}, {37, 3008}, {57, 347}, {81, 553}, {141, 519}, {192, 2325}, {239, 1654}, {284, 1429}, {516, 1386}, {536, 3589}, {740, 1125}, {894, 1266}, {950, 3100}, {1086, 1100}, {1323, 1418}, {2324, 2999}, {3618, 3729}, {3634, 3773}

### X(3947) = X(3601)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(a + b - c)(a - b + c)(a + 3b + 3c)

X(3947) lies on these lines: {1, 3091}, {2, 3361}, {7, 1698}, {10, 12}, {57, 3634}, {142, 1329}, {227, 3743}, {354, 3614}, {388, 1125}, {495, 946}, {516, 3085}, {519, 3485}, {551, 1388}, {1089, 1441}, {1478, 3612}, {1770, 3584}, {2099, 3625}, {3090, 3333}, {3340, 3626}, {3476, 3636}, {3600, 3624}

### X(3948) = X(3009)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c)(bc - a2)

X(3948) lies on these lines: {1, 3765}, {2, 39}, {10, 321}, {37, 313}, {75, 1213}, {86, 3770}, {92, 429}, {190, 2245}, {192, 2092}, {239, 350}, {242, 862}, {257, 312}, {314, 1654}, {329, 2899}, {341, 1834}, {344, 1234}, {514, 661}, {536, 3264}, {714, 3122}, {730, 3009}, {1269, 3739}

X(3948) = isogonal conjugate of X(18268)

### X(3949) = X(2260)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c)2(b2 + c2 - a2)

X(3949) lies on these lines: {6, 976}, {8, 1953}, {9, 943}, {10, 2294}, {12, 594}, {19, 200}, {37, 42}, {48, 78}, {63, 1796}, {69, 337}, {71, 72}, {100, 1761}, {198, 480}, {201, 2197}, {219, 1807}, {306, 3610}, {319, 1959}, {518, 2260}, {1089, 1826}, {1215, 2345}, {1757, 1778}

### X(3950) = X(9)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b + c - 3a)

X(3950) lies on these lines: {1, 346}, {6, 2325}, {8, 3731}, {9, 519}, {10, 37}, {45, 3625}, {142, 536}, {145, 1743}, {192, 3662}, {226, 3175}, {344, 3008}, {391, 3632}, {573, 3208}, {966, 3626}, {1018, 1400}, {1125, 2345}, {1266, 3644}, {1449, 3635}, {2171, 3671}, {2262, 2802}, {3664, 3729}

### X(3951) = X(3303)com[INVERSE(n(TANGENTIAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(3b + 3c + a)(b2 + c2 - a2)

X(3951) lies on these lines: {1, 2308}, {3, 63}, {8, 144}, {40, 3681}, {46, 3678}, {69, 3710}, {77, 201}, {329, 3091}, {377, 527}, {518, 3303}, {612, 1046}, {908, 3090}, {936, 3218}, {942, 3305}, {960, 3304}, {976, 1707}, {2093, 3617}, {2292, 3751}, {2475, 3679}, {3419, 3627}

### X(3952) = X(244)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   (b + c)/(b - c) : (c + a)/(c - a) : (a + b)/(a - c)

X(3952) lies on these lines: {2, 38}, {8, 80}, {10, 3120}, {37, 3121}, {72, 3701}, {100, 190}, {101, 835}, {110, 645}, {192, 872}, {210, 321}, {312, 3681}, {329, 2835}, {643, 765}, {644, 1783}, {660, 799}, {668, 891}, {726, 899}, {908, 3006}, {1265, 3436}, {3159, 3293}

X(3952) = isogonal conjugate of X(3733)
X(3952) = isotomic conjugate of X(7192)
X(3952) = anticomplement of X(244)
X(3952) = trilinear pole of line X(10)X(37) (the tangent to Kiepert hyperbola at X(10), and the line of the degenerate cross-triangle of Gemini triangles 15 and 16)
X(3952) = X(523)-cross conjugate of X(10)
X(3952) = perspector of ABC and side-triangle of Gemini triangles 17 and 18
X(3952) = barycentric product of vertices of Gemini triangle 17
X(3952) = barycentric product of vertices of Gemini triangle 18
X(3952) = intersection, other than A, B, C, of {ABC, Gemini 17}-circumconic and {ABC, Gemini 18}-circumconic

### X(3953) = X(3216)com(INTOUCH TRIANGLE)

Barycentrics   a (b^3 + c^3 + a b^2 + a c^2 - 2 a b c) : :

X(3953) lies on these lines: {1, 3}, {8, 1739}, {10, 244}, {21, 3315}, {38, 1125}, {39, 3726}, {321, 596}, {474, 3242}, {496, 3782}, {518, 3216}, {551, 2292}, {595, 3218}, {614, 1724}, {758, 1201}, {984, 3624}, {1015, 3721}, {1421, 1935}, {1736, 3086}, {3293, 3555}

### X(3954) = X(2295)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(b2 + c2)

X(3954) lies on these lines: {1, 6}, {8, 3735}, {10, 762}, {32, 976}, {38, 39}, {76, 321}, {100, 755}, {141, 1930}, {228, 2156}, {257, 668}, {274, 335}, {519, 3727}, {758, 2295}, {1125, 3726}, {1500, 2292}, {1575, 3670}, {1655, 2895}, {1843, 3688}, {2238, 3678}

### X(3955) = X(1848)com[INVERSE(n(ORTHIC TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a2 + bc)(b2 + c2 - a2)

X(3955) lies on these lines: {1, 987}, {3, 73}, {36, 1401}, {57, 182}, {63, 184}, {72, 1437}, {110, 3219}, {171, 2330}, {219, 3167}, {228, 295}, {293, 1214}, {394, 3781}, {436, 1948}, {511, 2003}, {611, 1460}, {982, 1428}, {1409, 2359}, {1707, 2175}, {3292, 3690}

X(3955) = crosssum of the polar conjugates of PU(6)

### X(3956) = X(549)com[EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(b2 + c2 - a2 + 5bc)

X(3956) lies on these lines: {10, 12}, {519, 3740}, {1962, 3293}, {2802, 3679}, {3214, 3743}, {3634, 3742}

### X(3957) = X(3748)com[INVERSE(n(INTOUCH TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 + b2 + c2 - 2ab - 2ac - bc)

X(3957) lies on these lines: {1, 2}, {21, 3555}, {38, 3750}, {55, 3218}, {63, 3243}, {81, 643}, {100, 354}, {149, 226}, {171, 3722}, {518, 1621}, {744, 2667}, {982, 2177}, {1001, 3681}, {1100, 3693}, {3158, 3306}, {3315, 3752}, {3434, 3475}, {3689, 3742}

### X(3958) = X(2294)com[INVERSE(n(EXTANGENTS TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(2a + b + c)(b2 + c2 - a2)

X(3958) lies on these lines: {1, 1778}, {6, 2292}, {9, 758}, {37, 2650}, {48, 63}, {71, 72}, {191, 284}, {201, 1409}, {219, 3157}, {896, 1333}, {960, 2260}, {1046, 2303}, {1100, 1962}, {1213, 3649}, {1449, 3743}, {1761, 2173}, {1839, 3686}, {1858, 2269}

### X(3959) = X(2176)com(ORTHIC TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b3 + c3 - b2c - bc2 + abc)

X(3959) lies on these lines: {1, 1929}, {2, 3727}, {6, 19}, {8, 3721}, {10, 3735}, {37, 3208}, {75, 257}, {85, 1086}, {145, 3726}, {517, 2176}, {982, 2319}, {986, 1107}, {1575, 3061}, {1738, 3094}, {1953, 2277}, {2170, 2275}, {2643, 3764}, {3057, 3290}

### X(3960) = X(676)com(HEXYL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b - c)(b2 + c2 - a2 - bc)

X(3960) lies on these lines: {1, 2254}, {2, 3762}, {3, 2814}, {88, 1022}, {101, 651}, {104, 106}, {214, 3738}, {241, 514}, {513, 1960}, {659, 764}, {667, 3777}, {676, 2826}, {690, 3743}, {812, 1015}, {830, 2530}, {900, 1387}, {1125, 3716}, {2006, 2401}

### X(3961) = X(171)com[INVERSE(n(INTOUCH TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 + b2 + c2 - ab - ac + bc)

X(3961) lies on these lines: {1, 2}, {9, 983}, {31, 1757}, {33, 242}, {37, 3684}, {38, 100}, {55, 846}, {63, 3099}, {171, 518}, {210, 238}, {595, 3678}, {668, 1965}, {756, 1621}, {902, 3219}, {982, 1054}, {1279, 3740}, {1763, 3465}, {3666, 3689}

### X(3962) = X(3146)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(3b2 + 3c2 - 3a2 - 2bc)

X(3962) lies on these lines: {1, 3683}, {8, 1836}, {10, 12}, {37, 2650}, {40, 3689}, {63, 2646}, {78, 1155}, {144, 145}, {329, 1837}, {354, 960}, {382, 517}, {392, 3636}, {942, 3624}, {1042, 2318}, {1464, 3682}, {1706, 3711}, {3555, 3635}

### X(3963) = X(2309)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c)(a2 + bc)

X(3963) lies on these lines: {2, 1240}, {9, 3765}, {10, 3728}, {37, 313}, {75, 141}, {76, 192}, {190, 3770}, {226, 306}, {495, 3695}, {536, 1269}, {668, 1654}, {730, 2309}, {732, 894}, {1018, 3729}, {1089, 3178}, {1920, 1926}, {3264, 3739}

### X(3964) = X(1609)com[INVERSE(n(TANGENTIAL TRIANGLE))]

Trilinears       t(A,B,C) : t(B,C,A) : t(C,A,B), where t(A,B,C) = cot2A cos A    (Peter Moses, 1/13/2012)
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b2 - c2 - a2)3

X(3964) lies on these lines: {3, 69}, {25, 317}, {95, 183}, {99, 1294}, {159, 1634}, {253, 2071}, {264, 1105}, {326, 1259}, {340, 3515}, {394, 577}, {491, 1584}, {492, 1583}, {524, 1609}, {648, 1033}, {1073, 2063}, {1270, 1599}, {1271, 1600}

X(3964) = isogonal conjugate of X(6524)
X(3964) = isotomic conjugate of X(1093)
X(3964) = X(92)-isoconjugate of X(2207)

### X(3965) = X(1400)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)2(b2 + c2 + ab + ac)

X(3965) lies on these lines: {6, 78}, {8, 37}, {9, 55}, {44, 2220}, {69, 241}, {72, 573}, {220, 3692}, {341, 346}, {391, 1212}, {518, 1400}, {872, 2340}, {960, 2269}, {1098, 1792}, {1211, 2092}, {1376, 2285}, {3057, 3169}, {3290, 3705}

### X(3966) = X(58)com[INVERSE(n(TANGENTIAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b + c + 2a)(b2 + c2 - a2)

X(3966) lies on these lines: {1, 1211}, {2, 1386}, {8, 210}, {9, 3703}, {38, 3764}, {55, 3687}, {69, 354}, {75, 1836}, {141, 614}, {219, 3686}, {306, 1001}, {333, 2194}, {345, 3683}, {391, 2348}, {958, 1036}, {3434, 3696}, {3715, 3717}

### X(3967) = X(3752)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(a2 - ab - ac + 2bc)

X(3967) lies on these lines: {12, 3710}, {37, 714}, {42, 3175}, {43, 536}, {65, 3701}, {72, 1089}, {75, 3740}, {210, 321}, {312, 518}, {329, 3416}, {726, 3752}, {908, 3703}, {1376, 3729}, {1901, 3610}, {2886, 3717}, {3681, 3706}

### X(3968) = X(549)com[INVERSE(n(HEXYL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(b2 + c2 - a2 - 7ab)

X(3968) lies on these lines: {2, 2802}, {10, 12}, {517, 547}, {519, 3742}

### X(3969) = X(2308)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b2 + c2 - a2 + bc)

X(3969) lies on these lines: {2, 594}, {8, 405}, {10, 1962}, {42, 3773}, {63, 544}, {81, 2295}, {100, 199}, {190, 2895}, {226, 306}, {319, 3219}, {345, 1150}, {502, 1089}, {961, 3476}, {1500, 3661}, {3006, 3706}, {3681, 3690}

### X(3970) = X(1334)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(b2 + c2 - bc - ca - ab)

X(3970) lies on these lines: {1, 6}, {35, 3509}, {39, 3726}, {55, 1759}, {65, 1018}, {79, 2795}, {321, 1930}, {758, 1334}, {942, 3693}, {1500, 3721}, {2170, 3244}, {2171, 3671}, {2276, 3670}, {2294, 2321}, {3216, 3290}, {3496, 3746}

### X(3971) = X(2)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(bc - ca - ab)

X(3971) lies on these lines: {1, 979}, {2, 726}, {10, 321}, {37, 714}, {43, 192}, {171, 190}, {210, 740}, {312, 984}, {354, 537}, {519, 3681}, {536, 3740}, {894, 1961}, {1211, 3773}, {1757, 1999}, {2901, 3678}, {3263, 3663}

### X(3972) = X(3314)com(3rd BROCARD TRIANGLE)

Barycentrics   2a4 + b2c2 : 2b4 + c2a2 : 2c4 + a2b2

X(3972) = eigencenter of 3rd Brocard triangle (Peter Moses, January 24, 2012); see X(3734)

X(3972) lies on these lines: {2, 187}, {3, 83}, {6, 99}, {32, 76}, {39, 3552}, {69, 1285}, {112, 264}, {183, 1384}, {251, 305}, {350, 609}, {376, 3618}, {574, 3329}, {691, 1316}, {754, 3314}, {1078, 3053}

### X(3973) = X(3161)com(EXCENTRAL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(3b +3c - 5a)

X(3973) lies on these lines: {1, 6}, {144, 3008}, {165, 2348}, {173, 363}, {200, 902}, {346, 3632}, {391, 3679}, {484, 2270}, {519, 3161}, {572, 3217}, {610, 2265}, {2163, 2297}, {2345, 3707}, {2347, 3730}, {2999, 3219}

### X(3974) = X(2999)com(EXCENTRAL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(a2 + b2 + c2 + 2bc)

X(3974) lies on these lines: {2, 1390}, {4, 1089}, {8, 210}, {33, 200}, {55, 346}, {321, 2550}, {329, 3416}, {344, 3757}, {391, 3715}, {612, 2345}, {756, 966}, {1219, 3304}, {3085, 3695}, {3161, 3683}, {3474, 3729}

### X(3975) = X(2664)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c - a)(a2 - bc)

X(3975) lies on these lines: {2, 330}, {8, 210}, {9, 3596}, {75, 966}, {190, 2183}, {239, 350}, {314, 3686}, {333, 3691}, {645, 2323}, {646, 2325}, {730, 2664}, {1655, 3666}, {1999, 3780}, {2340, 3699}, {3739, 3770}

### X(3976) = X(978)com(INTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b3 + c3 + ab2 + ac2 - 3abc)

X(3976) lies on these lines: {1, 3}, {8, 244}, {38, 3616}, {43, 3555}, {256, 3296}, {518, 978}, {749, 984}, {1015, 3061}, {1046, 1191}, {1329, 3756}, {1393, 3476}, {1739, 3632}, {2275, 3726}, {2292, 3622}, {2975, 3315}

### X(3977) = X(902)com[INVERSE(n(TANGENTIAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - 2a)(b2 + c2 - a2)

X(3977) lies on these lines: {2, 2415}, {3, 3710}, {63, 69}, {100, 2751}, {110, 2760}, {190, 908}, {344, 3306}, {441, 525}, {516, 3006}, {518, 3712}, {519, 902}, {726, 3011}, {1150, 2321}, {3219, 3687}, {3589, 3666}

X(3977) = isogonal conjugate of X(8752)
X(3977) = isotomic conjugate of X(6336)
X(3977) = crossdifference of every pair of points on line X(25)X(8643)

### X(3978) = X(694)com[INVERSE(n(1st BROCARD TRIANGLE))]

Barycentrics    b2c2(a2 - bc)(a2 + bc) : :
Barycentrics    directed distance of A to line PU(1) : :

X(3978) lies on these lines: {2, 39}, {6, 706}, {75, 256}, {83, 1207}, {99, 237}, {290, 325}, {308, 3589}, {315, 1899}, {316, 512}, {524, 670}, {561, 3765}, {671, 886}, {694, 698}, {702, 1084}, {1215, 1237}

X(3978) = isotomic conjugate of X(694)
X(3978) = anticomplement of X(3229)
X(3978) = perspector of conic {{A,B,C,PU(11)}}
X(3978) = trilinear pole of line PU(1) of 1st anti-Brocard triangle
X(3978) = trilinear pole of PU(133) (line X(804)X(5976))

### X(3979) = X(3750)com[INVERSE(n(INTOUCH TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 + b2 + c2 - 3ab - 3ac - bc)

X(3979) lies on these lines: {1, 2}, {81, 3722}, {238, 3748}, {354, 1054}, {518, 846}, {1046, 3746}, {1051, 1386}, {1621, 1757}

### X(3980) = X(612)com(1st BROCARD TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3 + b2c + bc2 + 2abc

X(3980) lies on these lines: {1, 3210}, {2, 846}, {10, 46}, {43, 894}, {57, 3741}, {75, 171}, {192, 1961}, {321, 750}, {612, 726}, {740, 940}, {968, 1125}, {986, 1010}, {1215, 1376}, {2345, 3509}, {3550, 3757}

### X(3981) = X(305)com(1st BROCARD TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b4 + c4 - b2c2)

X(3981) lies on these lines: {2, 694}, {4, 695}, {6, 25}, {22, 1691}, {23, 1501}, {305, 698}, {386, 2653}, {493, 1584}, {494, 1583}, {511, 1196}, {1350, 1611}, {1993, 2056}, {2979, 3231}, {3051, 3060}, {3095, 3117}

X(3981) = X(305)-of-1st-Brocard-triangle

### X(3982) = X(3683)com(INTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a + b - c)(a - b + c)(2a + 3b + 3c)

X(3982) lies on these lines: {1, 3529}, {2, 7}, {65, 3626}, {382, 950}, {388, 3632}, {516, 3748}, {546, 942}, {554, 3639}, {1081, 3638}, {1319, 3636}, {1374, 1659}, {2099, 3244}, {3487, 3528}, {3664, 3782}

### X(3983) = X(3523)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(b2 + c2 - a2 + 6bc)

X(3983) lies on these lines: {1, 3711}, {8, 3740}, {10, 12}, {37, 3214}, {40, 3715}, {354, 1698}, {392, 3626}, {405, 3689}, {518, 3619}, {936, 1319}, {960, 3617}, {3057, 3679}, {3555, 3634}, {3696, 3701}

### X(3984) = X(3304)com[INVERSE(n(TANGENTIAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(3b + 3c - a))(b2 + c2 - a2)

X(3984) lies on these lines: {1, 748}, {3, 63}, {8, 908}, {306, 1265}, {329, 3146}, {518, 3304}, {519, 2478}, {546, 3419}, {936, 3306}, {960, 3303}, {2476, 3679}, {3219, 3601}, {3243, 3622}, {3340, 3617}

### X(3985) = X(3290)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b + c - a) (bc - a2)

X(3985) lies on these lines: {9, 312}, {37, 714}, {190, 3509}, {210, 2321}, {341, 3208}, {522, 650}, {537, 3726}, {726, 3290}, {740, 2238}, {1089, 3294}, {1334, 3701}, {3684, 3685}, {3686, 3706}, {3691, 3702}

### X(3986) = X(3247)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(5a + b + c)

X(3986) lies on these lines: {1, 391}, {2, 2415}, {6, 551}, {9, 1125}, {10, 37}, {346, 1698}, {405, 1696}, {519, 966}, {1100, 3707}, {1400, 3294}, {1449, 3636}, {1743, 3616}, {2345, 3634}, {3244, 3686}

### X(3987) = X(3216)com[INVERSE(n(HEXYL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(b2 + c2 + ab + ac - 3bc)

X(3987) lies on these lines: {1, 474}, {8, 3670}, {10, 321}, {38, 3626}, {40, 1724}, {42, 3754}, {65, 3293}, {244, 3244}, {517, 3216}, {758, 3214}, {982, 3632}, {986, 3679}, {1201, 2802}, {1574, 3727}

X(3987) = crossdifference of every pair of points on line X(644)X(1783)

### X(3988) = X(546)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(3b2 + 3c2 - 3a2 + bc)

X(3988) lies on these lines: {10, 12}, {518, 3636}, {960, 3635}, {3632, 3681}

### X(3989) = X(1962)com(INCENTRAL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2b2 + 2c2 + ab + ac + 2bc)

X(3989) lies on these lines: {1, 2308}, {2, 726}, {37, 38}, {42, 984}, {45, 748}, {165, 612}, {517, 2292}, {518, 1962}, {614, 3731}, {756, 899}, {846, 902}, {896, 3745}, {1051, 1757}, {1961, 3218}

### X(3990) = X(1841)com[INVERSE(n(ORTHIC TRIANGLE))]

Barycentrics    a3(b + c)(b2 + c2 - a2)2 : :
Trilinears    (b + c) cos^2 A

X(3990) lies on these lines: {1, 6}, {48, 184}, {71, 73}, {255, 577}, {275, 321}, {287, 336}, {326, 394}, {517, 1841}, {604, 2198}, {836, 3682}, {919, 2749}, {1331, 2327}, {2269, 2288}, {3197, 3198}

X(3990) = X(92)-isoconjugate of X(28)

### X(3991) = X(1212)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(a2 + b2 + c2 - 2ab - 2ac)

X(3991) lies on these lines: {1, 728}, {9, 3295}, {10, 37}, {44, 2241}, {65, 1018}, {72, 1334}, {192, 3673}, {210, 3294}, {517, 3208}, {518, 3730}, {519, 1212}, {672, 3555}, {942, 3501}, {3509, 3579}

### X(3992) = X(1149)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c)(b + c - 2a)

X(3992) lies on these lines: {1, 341}, {10, 321}, {100, 2758}, {190, 484}, {312, 3679}, {523, 1577}, {726, 1739}, {1111, 3263}, {1479, 2899}, {1737, 3717}, {2901, 3214}, {3293, 3725}, {3626, 3702}, {3697, 3714}

### X(3993) = X(984)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(2a2 + ab + ac - bc)

X(3993) lies on these lines: {1, 87}, {10, 37}, {75, 1125}, {226, 1365}, {321, 1962}, {335, 2796}, {518, 3244}, {519, 751}, {536, 551}, {846, 1999}, {1215, 3175}, {1278, 3616}, {2667, 3159}, {3636, 3644}

### X(3994) = X(899)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(ab + ac - 2bc)

X(3994) lies on these lines: {10, 321}, {37, 2229}, {38, 312}, {42, 3175}, {190, 896}, {244, 726}, {523, 661}, {536, 899}, {750, 3729}, {1215, 1962}, {1266, 3263}, {2325, 3011}, {3632, 3681}, {3685, 3722}

### X(3995) = X(756)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(a2 + ab + ac - bc)

X(3995) lies on these lines: {1, 3159}, {2, 37}, {8, 2901}, {9, 3187}, {72, 145}, {81, 190}, {86, 1255}, {239, 3294}, {726, 3720}, {740, 756}, {1089, 3743}, {1215, 1962}, {1655, 2895}, {1999, 3219}

X(3995) = complement of X(4359)

### X(3996) = X(181)com[INVERSE(n(INTOUCH TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(a2 - ab - ac - bc)

X(3996) lies on these lines: {8, 21}, {10, 3750}, {145, 940}, {171, 519}, {190, 3681}, {200, 312}, {210, 3685}, {239, 3744}, {1222, 3621}, {2321, 3684}, {3210, 3242}, {3550, 3632}, {3689, 3706}, {3696, 3757}

### X(3997) = X(3735)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(2a2 + bc)

X(3997) lies on these lines: {1, 672}, {6, 519}, {10, 213}, {37, 758}, {42, 1018}, {44, 1573}, {58, 2329}, {81, 644}, {101, 171}, {386, 3501}, {551, 3230}, {1125, 2176}, {3625, 3780}

### X(3998) = X(1779)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(b2 + c2 - a2)2

X(3998) lies on these lines: {2, 37}, {3, 63}, {81, 1257}, {100, 1297}, {213, 2221}, {306, 307}, {326, 394}, {518, 2352}, {1073, 3692}, {1108, 3187}, {1210, 2901}, {1331, 1801}, {2287, 3219}

X(3998) = isogonal conjugate of X(5317)

### X(3999) = X(899)com(INTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(3b2 + 3c2 - ab - ac - 4bc)

X(3999) lies on these lines: {1, 3}, {38, 3742}, {44, 3290}, {88, 1280}, {244, 518}, {896, 3246}, {908, 3756}, {1054, 3689}, {1279, 3218}, {3240, 3752}, {3242, 3306}, {3667, 3676}, {3693, 3726}

### X(4000) = X(346)com[T(-a, c)]

Barycentrics    a2 + b2 + c2 - 2bc : :
Barycentrics    cot^2(B/2) + cot^2(C/2) : :
Barycentrics    b c - SW : :

As described in the preamble to X(3758), the notation T(-a, c) represents the central triangle whose A-vertex is the point -a : c : b. This triangle and seven others are discussed just before X(4357).

X(4000) is the center of the inellipse that is the barycentric square of the Gergonne line. The Brianchon point (perspector) of the inellipse is X(279). (Randy Hutson, October 15, 2018)

X(4000) lies on these lines: {1, 142}, {2, 37}, {3, 1612}, {4, 990}, {6, 7}, {8, 141}, {9, 3008}, {10, 4353}, {19, 57}, {20, 1104}, {27, 2221}, {31, 3474}, {42, 3475}, {44, 144}, {48, 1429}, {69, 239}, {77, 3554}, {105, 1486}, {145, 3834}, {172, 4209}, {193, 320}, {226, 2999}, {241, 347}, {273, 393}, {279, 1418}, {319, 3620}, {329, 3782}, {348, 2275}, {386, 2140}, {387, 942}, {388, 4327}, {390, 1279}, {497, 614}, {499, 1733}, {518, 4310}, {527, 1743}, {594, 3763}, {894, 3618}, {910, 3598}, {938, 1834}, {962, 1191}, {1100, 3945}, {1107, 4352}, {1125, 4356}, {1193, 3485}, {1210, 1861}, {1266, 3729}, {1386, 4307}, {1423, 2183}, {1449, 3664}, {1453, 1890}, {1471, 4331}, {1473, 1851}, {1618, 2175}, {1714, 3670}, {1722, 2551}, {1760, 3218}, {3086, 4008}, {3212, 3959}, {3247, 4021}, {3486, 3924}, {3619, 3661}, {3875, 3912}

X(4000) = isogonal conjugate of X(7123)
X(4000) = complement of X(346)
X(4000) = {X(2),X(75)}-harmonic conjugate of X(2345)
X(4000) = isotomic conjugate of X(30701)
X(4000) = anticomplement of X(17279)

### X(4001) = X(42)com[INVERSE(n(TANGENTIAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (2a + b + c)(b2 + c2 - a2)

X(4001) lies on these lines: {2, 1743}, {8, 2093}, {63, 69}, {222, 307}, {226, 1150}, {320, 333}, {321, 527}, {524, 3666}, {553, 3578}, {1125, 2308}, {1269, 1839}, {2895, 3218}, {3187, 3663}

### X(4002) = X(3523)com[INVERSE(n(HEXYL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(b2 + c2 - a2 - 8bc)

X(4002) lies on these lines: {10, 12}, {354, 3626}, {392, 1698}, {405, 1706}, {474, 1420}, {517, 3090}, {942, 3617}, {956, 3361}, {1376, 3612}, {3057, 3634}, {3555, 3679}, {3632, 3742}

### X(4003) = X(3240)com(INTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(3b2 + 3c2 + ab + ac - 2bc)

X(4003) lies on these lines: {1, 3}, {11, 3663}, {37, 244}, {38, 210}, {45, 3290}, {88, 1390}, {518, 3240}, {596, 3634}, {614, 3683}, {1386, 3218}, {3210, 3706}, {3242, 3689}

### X(4004) = X(3091)com[INVERSE(n(HEXYL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(3b2 + 3c2 - 3a2 - 8bc)

X(4004) lies on these lines: {10, 12}, {78, 1159}, {145, 942}, {354, 3635}, {392, 3624}, {405, 2093}, {474, 3340}, {517, 631}, {956, 3339}, {1482, 3306}, {3057, 3636}, {3555, 3632}

### X(4005) = X(3091)com(EXTOUCH)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(3b2 + 3c2 - 3a2 + 2bc)

X(4005) lies on these lines: {1, 3715}, {10, 12}, {40, 3711}, {44, 976}, {145, 960}, {354, 3624}, {392, 3635}, {518, 3616}, {1898, 3059}, {2318, 2594}, {3057, 3632}, {3555, 3636}

### X(4006) = X(1475)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(b2 + c2 - ab - ac + bc)

X(4006) lies on these lines: {9, 3746}, {25, 200}, {37, 762}, {72, 1018}, {210, 3294}, {442, 594}, {1089, 1826}, {1334, 3678}, {1574, 3726}, {2170, 3625}, {3061, 3632}, {3681, 3730}

### X(4007) = X(1449)com(EXTOUCH TRIANGLE)

Barycentrics   (b + c - a)(2b + 2c + a) : :

X(4007) lies on these lines: {1, 594}, {6, 3632}, {8, 9}, {10, 3247}, {37, 3679}, {319, 3729}, {519, 1449}, {966, 3626}, {1100, 3633}, {1266, 3620}, {2323, 3713}, {3624, 3723}

X(4007) = complement of X(4460)

### X(4008) = X(1423)com[INVERSE(n(HEXYL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(3a4 + b4 + c4 - 2b2c2)

X(4008) lies on these lines: {1, 75}, {19, 158}, {31, 92}, {82, 91}, {239, 613}, {240, 774}, {560, 1955}, {611, 894}, {920, 1760}, {1210, 1738}, {2345, 3085}, {2550, 3419}

Let A'B'C' be the Artzt triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(4008). (Randy Hutson, April 9, 2016)

### X(4009) = X(899)com[INVERSE(n(HEXYL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(ab + ac - 2bc)

X(4009) lies on these lines: {8, 210}, {11, 3717}, {43, 3175}, {190, 1155}, {321, 3740}, {522, 650}, {536, 899}, {986, 1698}, {1125, 1215}, {1329, 3710}, {3452, 3703}, {3685, 3689}

### X(4010) = X(659)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 - c2)(a2 - bc)

X(4010) lies on these lines: {1, 2787}, {11, 244}, {149, 3448}, {320, 350}, {512, 1577}, {522, 1491}, {523, 661}, {659, 812}, {663, 814}, {804, 3027}, {891, 3762}, {1639, 2977}

X(4010) = isotomic conjugate of X(4589)
X(4010) = crossdifference of every pair of points on line X(58)X(101)

### X(4011) = X(614)com(1st BROCARD TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a3 + b2c + bc2 - 2abc)

X(4011) lies on these lines: {1, 979}, {2, 846}, {9, 3741}, {10, 1479}, {43, 3685}, {190, 982}, {238, 312}, {321, 748}, {614, 726}, {908, 3771}, {997, 3465}, {1001, 1215}

### X(4012) = X(269)com(EXTOUCH TRIANGLE)

Barycentrics    (b + c - a)3(a2 + b2 + c2 - 2bc) : :

Let DEF be the extouch triangle of ABC. Let Ha be the hyperbola with foci E and F, passing through A, and define Hb and Hc cyclically. These three hyperbolas have two common points, U and V. Let Pa be the pole of the line UV with respect to Ha, and define Pb and Pc cyclically. Then DEF and PaPBPc are perspective at X(4012). (Angel Montesdeoca, September 23, 2018)

X(4012) lies on these lines: {7, 8}, {33, 200}, {346, 480}, {2823, 3421}

### X(4013) = X(214)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)2/(b + c - 2a)

X(4013) lies on these lines: {10, 3120}, {12, 1365}, {80, 519}, {88, 1224}, {106, 1125}, {115, 594}, {121, 1086}, {502, 3178}, {596, 1329}, {901, 2372}, {903, 1268}, {1089, 1109}

### X(4014) = X(190)com(INTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b - c)2(a2 - ab - ac + 2bc)

X(4014) lies on these lines: {7, 2481}, {11, 1357}, {65, 2801}, {513, 1086}, {516, 1463}, {1015, 3123}, {1356, 3026}, {1358, 2820}, {1364, 1365}, {1401, 1836}, {2223, 3000}, {2310, 3020}

### X(4015) = X(140)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(b2 + c2 - a2 + 3bc)

X(4015) lies on these lines: {10, 12}, {100, 3065}, {392, 3625}, {405, 3711}, {518, 3634}, {756, 3293}, {762, 2238}, {936, 1476}, {960, 2802}, {1125, 3740}, {1126, 1961}, {1698, 3681}

### X(4016) = X(6)com(INCENTRAL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(b3 + c3 + ab2 + ac2)

X(4016) lies on these lines: {1, 1333}, {6, 758}, {37, 65}, {38, 1755}, {257, 3770}, {740, 3416}, {902, 1962}, {1100, 2650}, {1213, 3125}, {2220, 3496}, {2305, 3743}, {2643, 3728}

### X(4017) = X(3737)com(INTOUCH TRIANGLE)

Trilinears    sin A sin(B - C) tan(A/2) : :
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 - c2)(a + b - c)(a - b + c)

X(4017) lies on these lines: {56, 3733}, {73, 3657}, {109, 1290}, {244, 1365}, {513, 663}, {522, 693}, {523, 656}, {647, 661}, {798, 1400}, {934, 2701}, {3020, 3123}

X(4017) = isogonal conjugate of X(643)
X(4017) = crossdifference of every pair of points on line X(9)X(21)

### X(4018) = X(3529)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(3b2 + 3c2 - 3a2 - 4bc)

X(4018) lies on these lines: {1, 3052}, {10, 12}, {20, 145}, {354, 3636}, {392, 942}, {474, 3339}, {518, 3632}, {956, 3340}, {960, 3624}, {1385, 3218}, {3057, 3635}

### X(4019) = X(1964)com[INVERSE(n(TANGENTIAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(a2 + bc)(b2 + c2 - a2)

X(4019) lies on these lines: {46, 1089}, {63, 3718}, {69, 337}, {171, 385}, {190, 1761}, {192, 986}, {306, 307}, {321, 1400}, {740, 3778}, {813, 2868}, {1237, 1840}

### X(4020) = X(1840)com[INVERSE(n(ORTHIC TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(b2 + c2)(b2 + c2 - a2)

X(4020) lies on these lines: {1, 1755}, {3, 295}, {39, 1401}, {48, 255}, {63, 304}, {799, 1925}, {922, 1917}, {1475, 2225}, {1496, 1973}, {1923, 1964}, {1930, 3404}

### X(4021) = X(1100)com(INTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b2 + c2 - 2a2 - 3ab - 3ac + 2bc

X(4021) lies on these lines: {1, 7}, {37, 3008}, {69, 3244}, {75, 1125}, {86, 1266}, {319, 519}, {527, 1100}, {1086, 3723}, {1268, 3634}, {1387, 2783}, {2325, 3589}

### X(4022) = X(872)com(INCENTRAL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(ab3 + ac3 + b3c + bc3)

X(4022) lies on these lines: {1, 1918}, {37, 38}, {69, 3764}, {75, 982}, {141, 2228}, {244, 3728}, {256, 320}, {518, 872}, {740, 3670}, {749, 984}, {2667, 3666}

### X(4023) = X(750)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b2 + c2 + 2ab + 2ac)

X(4023) lies on these lines: {8, 11}, {9, 3712}, {43, 1211}, {141, 899}, {210, 3687}, {306, 3740}, {345, 3715}, {524, 750}, {908, 3696}, {1150, 3035}, {3452, 3706}

### X(4024) = X(661)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics    (b + c)(b2 - c2) : :

X(4024) lies on the Yff parabola and these lines: {37, 650}, {42, 2395}, {101, 476}, {190, 892}, {321, 693}, {522, 649}, {523, 661}, {657, 3064}, {685, 1897}, {784, 3250}, {850, 1577}

X(4024) = isogonal conjugate of X(4556)
X(4024) = isotomic conjugate of X(4610)
X(4024) = trilinear pole of line X(115)X(2643)
X(4024) = barycentric product X(i)*X(j) for these {i,j}: {10,523}, {514,594}

### X(4025) = X(649)com[INVERSE(n(TANGENTIAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)(b2 + c2 - a2)

X(4025) lies on these lines: {2, 2400}, {63, 652}, {239, 514}, {441, 525}, {513, 3004}, {521,4131}, {522, 693}, {650, 918}, {658, 1897}, {812, 3776}, {934, 2765}, {1638, 3700}

X(4025) = isogonal conjugate of X(8750)
X(4025) = isotomic conjugate of X(1897)
X(4025) = anticomplement of X(3239)
X(4025) = pole of Soddy line wrt Steiner circumellipse
X(4025) = crossdifference of every pair of points on line X(25)X(41)

### X(4026) = X(572)com[INVERSE(n(HEXYL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b2 + c2 + 2a2 + ab + ac)

X(4026) lies on these lines: {1, 141}, {2, 11}, {10, 37}, {12, 1284}, {42, 1211}, {65, 307}, {238, 3589}, {405, 1486}, {519, 3775}, {1125, 1279}, {1738, 3739}

X(4026) = barycentric square of X(385)
X(4026) = barycentric product X(4366)*X(6645)
X(4026) = barycentric square of X(385)
X(4026) = barycentric product X(4366)*X(6645)

### X(4027) = X(384)com[INVERSE(n(1st BROCARD TRIANGLE))]

Barycentrics    (a4 - b2c2)2

X(4027) lies on these lines: {2, 98}, {6, 1916}, {32, 99}, {83, 115}, {239, 1281}, {384, 2782}, {385, 732}, {620, 1078}, {733, 1084}, {2023, 3329}, {3044, 3203}

X(4027) = 6th-Brocard-to-ABC similarity image of X(384)
X(4027) = homothetic center of 1st and 6th anti-Brocard triangles
X(4027) = perspector of ABC and 1st Brocard triangle of 6th anti-Brocard triangle
X(4027) = perspector of ABC and cross-triangle of ABC and 1st anti-Brocard triangle
X(4027) = endo-homothetic center of 1st and 6th Brocard triangles

### X(4028) = X(63)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b2 + c2 - 3a2)

X(4028) lies on these lines: {1, 2}, {101, 2374}, {171, 332}, {193, 1707}, {226, 740}, {345, 3751}, {430, 1867}, {1215, 2321}, {1869, 3189}, {2129, 2333}, {2887, 3755}

### X(4029) = X(45)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b + c - 5a)

X(4029) lies on these lines: {1, 2325}, {6, 3635}, {9, 145}, {10, 37}, {44, 3244}, {45, 519}, {142, 192}, {346, 3247}, {1449, 3161}, {2345, 3624}, {3632, 3686}

### X(4030) = X(38)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(2a2 + b2 + c2)

X(4030) lies on these lines: {8, 21}, {10, 3744}, {312, 3058}, {321, 528}, {519, 3666}, {594, 1914}, {3679, 3749}, {3683, 3717}, {3686, 3693}, {3687, 3689}, {3695, 3746}

### X(4031) = X(3711)com(INTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c + 4a)/(b + c - a)

X(4031) lies on these lines: {1, 3528}, {2, 7}, {56, 3636}, {65, 1317}, {546, 1210}, {550, 942}, {946, 1768}, {950, 3529}, {1000, 2093}, {3339, 3632}

### X(4032) = X(3688)com(INTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(a2 + bc)/(b + c - a)

X(4032) lies on these lines: {7, 192}, {37, 226}, {57, 75}, {65, 740}, {85, 1221}, {201, 388}, {304, 3729}, {536, 553}, {894, 2329}, {1400, 1441}

### X(4033) = X(3248)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c)/(b - c)

X(4033) is the intersection, other than the vertices of the Gemini triangle 17, of the {ABC, Gemini 17}-circumconic and {Gemini 17, Gemini 18}-circumconic. (Randy Hutson, November 30, 2018)

X(4033) lies on these lines: {10, 3122}, {75, 141}, {100, 835}, {101, 839}, {190, 646}, {313, 2321}, {341, 3695}, {645, 1016}, {1897, 3699}, {2295, 3758}

X(4033) = isotomic conjugate of X(1019)
X(4033) = trilinear pole of line X(10)X(321)

### X(4034) = X(3247)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(2b + 2c + 3a)

X(4034) lies on these lines: {1, 1213}, {6, 3679}, {8, 9}, {10, 1449}, {37, 3632}, {519, 966}, {553, 3646}, {594, 1743}, {1100, 1698}, {2345, 3626}

### X(4035) = X(3052)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(3b2 + 3c2 - 3a2 - 2bc)

X(4035) lies on these lines: {2, 1449}, {142, 3687}, {226, 306}, {329, 2325}, {345, 527}, {519, 3772}, {1427, 3694}, {2887, 3755}, {3061, 3452}, {3671, 3704}

### X(4036) = X(2605)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c)(b2 + c2)

X(4036) lies on these lines: {10, 522}, {37, 2395}, {100, 476}, {313, 3261}, {424, 2501}, {513, 2517}, {523, 1577}, {668, 892}, {685, 692}, {2787, 3733}

### X(4037) = X(2238)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)2(a2 - bc)

X(4037) lies on these lines: {2, 37}, {213, 2901}, {230, 3712}, {523, 661}, {594, 756}, {726, 3726}, {740, 2238}, {1089, 1500}, {1107, 3702}, {1914, 3685}

### X(4038) = X(1961)com(INTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 + 2ab + 2ac + 3bc)

X(4038) lies on these lines: {1, 3}, {2, 3775}, {81, 238}, {86, 3741}, {333, 1125}, {518, 1961}, {1100, 3684}, {1126, 3634}, {1206, 3231}, {1962, 3218}

### X(4039) = X(1959)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(a4 - b2c2)

X(4039) lies on these lines: {1, 2}, {31, 3765}, {171, 1909}, {313, 983}, {385, 1580}, {714, 2245}, {730, 2223}, {740, 1284}, {846, 1655}, {1215, 2295}

### X(4040) = X(1734)com(HEXYL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b - c)(a2 - ab - ac - bc)

X(4040) lies on these lines: {36, 238}, {41, 2141}, {512, 659}, {522, 3465}, {649, 2664}, {650, 1734}, {661, 830}, {1491, 3216}, {1577, 3716}

### X(4041) = X(1019)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(b2 - c2)

X(4041) lies on these lines: {10, 1577}, {42, 810}, {100, 2701}, {512, 661}, {514, 1734}, {522, 3717}, {523, 656}, {650, 663}, {667, 1635}, {891, 2530}

X(4041) = isogonal conjugate of X(1414)
X(4041) = isotomic conjugate of X(4625)
X(4041) = crossdifference of every pair of points on line X(57)X(77)

### X(4042) = X(968)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(a2 + 2bc + 2ca + 2ab)

X(4042) lies on these lines: {8, 21}, {9, 3706}, {10, 940}, {63, 3696}, {171, 3679}, {219, 3686}, {312, 3715}, {391, 497}, {1150, 1376}, {3632, 3750}

### X(4043) = X(872)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c)(a2 - bc - ca - ab)

X(4043) lies on these lines: {2, 37}, {190, 314}, {213, 3759}, {313, 2321}, {518, 3702}, {740, 872}, {984, 3159}, {1043, 3191}, {1215, 2667}, {3696, 3701}

### X(4044) = X(869)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c)(a2 - bc - ca - ab)

X(4044) lies on these lines: {2, 3760}, {10, 321}, {76, 85}, {142, 1269}, {306, 1230}, {307, 1229}, {313, 2321}, {519, 3765}, {1400, 3729}, {1500, 3175}

### X(4045) = X(141)com(1st BROCARD TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 + 2a2b2 + 2a2c2

X(4045) lies on these lines: {2, 99}, {6, 754}, {30, 3589}, {39, 325}, {141, 538}, {182, 2794}, {194, 3096}, {316, 3329}, {597, 2030}, {2679, 3111}

X(4045) = X(141)-of-1st-Brocard-triangle

### X(4046) = X(81)com(EXTOUCH TRIANGLE)

Barycentrics   (b + c)(b + c - a)(b + c + 2a) : :

X(4046) lies on these lines: {8, 21}, {11, 3687}, {42, 594}, {210, 2321}, {306, 3696}, {346, 3715}, {519, 3745}, {740, 1211}, {1213, 1962}, {3683, 3686}

### X(4047) = X(37)com[INVERSE(n(EXTANGENTS TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(b + c + 3a)(b2 + c2 - a2)

X(4047) lies on these lines: {9, 65}, {10, 1901}, {37, 758}, {46, 965}, {63, 77}, {71, 72}, {144, 1441}, {392, 2260}, {516, 3686}, {579, 960}

### X(4048) = X(32)com(1st BROCARD TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a6 + b4c2 + b2c4

X(4048) lies on these lines: {3, 66}, {6, 194}, {32, 732}, {69, 2076}, {76, 1691}, {99, 737}, {182, 2782}, {206, 3492}, {305, 1915}, {524, 1003}

### X(4049) = X(3251)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics    (b2 - c2 - a2)/(b + c - 2a) : :

X(4049) lies on the Kiepert hyperbola and these lines: {2, 514}, {4, 2457}, {10, 523}, {76, 3261}, {98, 106}, {321, 1577}, {671, 903}, {901, 2690}, {1797, 2986}

X(4049) = orthocenter of X(2)X(4)X(10)
X(4049) = barycentric product X(106)*X(850)
X(4049) = barycentric quotient X(106)/X(110)

### X(4050) = X(3212)com[INVERSE(n(INTOUCH TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(3bc + ab + ac)

X(4050) lies on these lines: {1, 1575}, {8, 9}, {40, 2784}, {519, 3501}, {672, 3621}, {1018, 3632}, {1449, 2295}, {2092, 3247}, {3625, 3730}

### X(4051) = X(3208)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(b2 + c2 - 3bc)

X(4051) lies on these lines: {1, 2271}, {8, 2170}, {9, 3057}, {956, 3496}, {982, 2319}, {984, 3727}, {1212, 3208}, {2082, 2329}, {2802, 3730}

### X(4052) = X(3158)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)/(b + c - 3a)

X(4052) lies on these lines: {2, 2415}, {4, 519}, {30, 3429}, {98, 1293}, {226, 3175}, {262, 726}, {516, 3424}, {536, 2051}, {551, 3445}

### X(4053) = X(2245)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)2(b2 + c2 - a2 - bc)

X(4053) lies on these lines: {1, 6}, {12, 594}, {321, 3262}, {523, 661}, {524, 1959}, {758, 2245}, {1030, 1761}, {1213, 2294}, {1259, 2178}

### X(4054) = X(2177)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b2 + c2 - a2 - 4bc)

X(4054) lies on these lines: {2, 2415}, {10, 3120}, {75, 908}, {226, 306}, {312, 1269}, {329, 391}, {442, 3710}, {527, 1150}, {3649, 3714}

### X(4055) = X(1860)com[INVERSE(n(ORTHIC TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4(b + c)(b2 + c2 - a2)2

X(4055) lies on these lines: {6, 31}, {10, 275}, {58, 1794}, {184, 2200}, {201, 2650}, {228, 1409}, {255, 394}, {287, 293}, {516, 1860}

### X(4056) = X(1759)com(INTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 - a4 - b3c - bc3

X(4056) lies on these lines: {4, 1111}, {7, 79}, {30, 3665}, {69, 1089}, {80, 3212}, {85, 3585}, {315, 1930}, {320, 3760}, {3583, 3673}

### X(4057) = X(1459)com(TANGENTIAL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b - c)(a2 + ab + ac - bc)

X(4057) lies on these lines: {3, 3667}, {6, 1919}, {23, 385}, {36, 238}, {522, 1324}, {663, 834}, {884, 3415}, {1960, 2605}, {2483, 3709}

X(4057) = isogonal conjugate of X(8050)
X(4057) = polar conjugate of isotomic conjugate of X(22154)
X(4057) = pole, with respect to circumcircle, of the Nagel line
X(4057) = crossdifference of every pair of points on line X(37)X(39)

### X(4058) = X(1449)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(3b + 3c -a)

X(4058) lies on these lines: {6, 3625}, {8, 1743}, {9, 3626}, {10, 37}, {346, 3679}, {519, 1449}, {1278, 3661}, {3247, 3634}, {3617, 3731}

### X(4059) = X(1334)com(INTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a + b - c)(a - b + c)(ab + ac + 2bc)

X(4059) lies on these lines: {7, 8}, {226, 241}, {279, 3485}, {354, 3673}, {942, 1111}, {950, 3664}, {1358, 2795}, {1434, 1447}, {3691, 3739}

### X(4060) = X(1100)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(3b + 3c + 2a)

X(4060) lies on these lines: {8, 9}, {37, 3626}, {319, 527}, {519, 594}, {1449, 3621}, {2345, 3632}, {3247, 3617}, {3634, 3723}

### X(4061) = X(940)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b + c - a)(3a + b + c)

X(4061) lies on these lines: {1, 2}, {55, 3686}, {209, 3059}, {210, 2321}, {226, 3696}, {1211, 3755}, {2325, 3715}, {3452, 3706}, {3683, 3707}

### X(4062) = X(896)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b2 + c2 - 2a2)

X(4062) lies on these lines: {1, 2}, {101, 2770}, {523, 661}, {524, 896}, {740, 3120}, {846, 2895}, {1211, 1962}, {2177, 3416}, {2650, 3704}

### X(4063) = X(663)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b - c)(a2 + ab + ac - bc)

X(4063) lies on these lines: {1, 667}, {40, 3309}, {57, 1022}, {239, 514}, {484, 513}, {512, 659}, {764, 3336}, {798, 812}, {834, 3737}

### X(4064) = X(656)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b2 - c2)(b2 + c2 - a2)

X(4064) lies on these lines: {10, 2394}, {37, 2509}, {71, 879}, {72, 521}, {101, 935}, {190, 2966}, {522, 3465}, {523, 661}, {525, 656}

### X(4065) = X(596)com(INCENTRAL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(2a + b + c)(a2 + ab + ac - bc)

X(4065) lies on these lines: {1, 596}, {10, 37}, {42, 3159}, {190, 1126}, {519, 2292}, {726, 2667}, {758, 3057}, {1125, 1962}, {2650, 3635}

X(4065) = reflection of X(4647) in X(1125)
X(4065) = complement of X(4647) wrt incentral triangle
X(4065) = anticomplement of X(1125) wrt incentral triangle
X(4065) = X(1)-Ceva conjugate of X(1125)

### X(4066) = X(386)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c)(2b + 2c + a)

X(4066) lies on these lines: {8, 3583}, {10, 321}, {75, 3634}, {312, 1125}, {758, 3714}, {1215, 2901}, {3175, 3743}, {3244, 3702}, {3454, 3773}

### X(4067) = X(382)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(2b2 + 2c2 - 2a2 - bc)

X(4067) lies on these lines: {1, 2308}, {8, 3585}, {10, 12}, {63, 3612}, {517, 3625}, {518, 3244}, {551, 960}, {997, 3361}, {3626, 3681}

### X(4068) = X(142)com(INCENTRAL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b + c)(a2 - ab - ac - 3bc)

X(4068) lies on these lines: {1, 3286}, {6, 2667}, {45, 3728}, {55, 199}, {523, 885}, {740, 1001}, {1486, 3295}, {2223, 3723}, {2292, 3242}

### X(4069) = X(3020)com[INVERSE(n(INTOUCH TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(b + c - a)/(b - c)

X(4069) lies on these lines: {1, 597}, {9, 3056}, {190, 1026}, {645, 3737}, {646, 3699}, {692, 1023}, {1296, 2748}, {2325, 2340}

### X(4070) = X(2243)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b3 + c3 - 2a3)

X(4070) lies on these lines: {8, 41}, {320, 3509}, {522, 650}, {752, 2243}, {1125, 3290}, {2246, 3006}, {2321, 3689}, {2348, 3707}

### X(4071) = X(1914)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b3 + c3 - a3 - abc)

X(4071) lies on these lines: {10, 213}, {37, 744}, {226, 306}, {334, 350}, {594, 1215}, {672, 3006}, {1281, 3509}, {1500, 3178}

### X(4072) = X(1743)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(3b + 3c - 5a)

X(4072) lies on these lines: {9, 3625}, {10, 37}, {346, 519}, {551, 2345}, {1449, 3244}, {3161, 3632}, {3626, 3731}, {3644, 3663}

### X(4073) = X(1423)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)2(b2 + c2 - bc)

X(4073) lies on these lines: {8, 192}, {9, 55}, {78, 238}, {346, 2310}, {518, 1423}, {982, 2887}, {3056, 3061}, {3190, 3725}

### X(4074) = X(1194)com(1st BROCARD TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 + c2)(a4 + b2c2)

X(4074) lies on these lines: {2, 694}, {6, 305}, {76, 1613}, {141, 427}, {384, 1915}, {698, 1194}, {732, 3051}, {1799, 2076}

### X(4075) = X(1125)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)2(a2 + ab + ac - bc)

X(4075) lies on these lines: {2, 596}, {10, 321}, {12, 1365}, {210, 2901}, {519, 960}, {726, 3634}, {1125, 1215}, {3175, 3697}

X(4075) = complement of X(596)
X(4075) = centroid of X(10) plus the vertices of the anticevian triangle of X(10)

### X(4076) = X(1052)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/(b - c)2

X(4076) lies on these lines: {99, 2748}, {190, 3667}, {238, 519}, {522, 3699}, {644, 3063}, {645, 3737}, {1447, 3263}, {2325, 3684}

X(4076) = perspector of ABC and the reflection of the intouch triangle in the Nagel line
X(4076) = isogonal conjugate of X(1357)
X(4076) = isotomic conjugate of X(1358)

### X(4077) = X(1021)com(INTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b2 - c2)/(b + c - a)

X(4077) lies on these lines: {109, 2864}, {226, 661}, {514, 3064}, {522, 693}, {525, 1577}, {934, 2689}, {1365, 3323}, {1367, 3326}

X(4077) = isotomic conjugate of X(643)

### X(4078) = X(1001)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b2 + c2 - a2 - 2ab - 2ac)

X(4078) lies on these lines: {1, 344}, {10, 37}, {45, 3416}, {142, 726}, {192, 1738}, {306, 756}, {519, 1001}, {1279, 3244}

### X(4079) = X(798)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b + c)(b2 - c2)

X(4079) lies on these lines: {37, 513}, {101, 691}, {213, 3063}, {512, 798}, {523, 661}, {663, 1919}, {786, 3766}, {1918, 2422}

X(4079) = isogonal conjugate of X(4610)

### X(4080) = X(678)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)/(b + c - 2a)

Let A14B14C14 be Gemini triangle 14. Let A' be the perspector of conic {{A,B,C,B14,C14}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(4080). (Randy Hutson, January 15, 2019)

X(4080) lies on these lines: {2, 45}, {4, 145}, {10, 3120}, {76, 1978}, {98, 901}, {106, 835}, {329, 1751}, {908, 1266}

X(4080) = isogonal conjugate of X(3285)
X(4080) = isotomic conjugate of X(16704)
X(4080) = complement of X(30579)
X(4080) = trilinear pole of line X(10)X(523)

### X(4081) = X(651)com(EXTOUCH TRIANGLE)

Trilinears    cot^2(A/2) sin^2(B/2 - C/2) : :
Barycentrics    (b - c)2(b + c - a)3 : :

X(4081) lies on these lines: {8, 190}, {11, 123}, {12, 318}, {56, 280}, {346, 480}, {653, 1360}, {1146, 2310}, {2321, 3059}

X(4081) = trilinear pole wrt extouch triangle of Nagel line
X(4081) = trilinear square of X(6730)

### X(4082) = X(614)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b + c - a)2

X(4082) lies on these lines: {10, 321}, {55, 2325}, {200, 346}, {210, 2321}, {312, 3717}, {3175, 3755}, {3452, 3703}, {3686, 3715}

### X(4083) = X(512)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b - c)(ab + ac - bc)

X(4083) lies on these lines: {1, 667}, {30, 511}, {65, 876}, {650, 3250}, {659, 663}, {693, 2533}, {1734, 2530}, {2254, 3777}

X(4083) = isogonal conjugate of X(932)
X(4083) = crossdifference of every pair of points on line X(6)X(43)

### X(4084) = X(382)com[INVERSE(n(HEXYL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(2b2 + 2c2 - 2a2 - 3bc)

X(4084) lies on these lines: {1, 89}, {10, 12}, {517, 550}, {518, 3625}, {551, 942}, {958, 1159}, {997, 3339}, {2802, 3555}

### X(4085) = X(182)com[INVERSE(n(HEXYL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b2 + c2 - 2a2 + bc)

X(4085) lies on these lines: {2, 2177}, {6, 752}, {8, 3775}, {10, 37}, {42, 2887}, {141, 519}, {528, 3589}, {537, 3663}

### X(4086) = X(3737)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b2 - c2)(b + c - a)

X(4086) lies on these lines: {10, 656}, {100, 2689}, {513, 3762}, {514, 2517}, {522, 3717}, {523, 1577}, {3064, 3239}

X(4086) = isotomic conjugate of X(1414)
X(4086) = X(57)-isoconjugate of X(163)

### X(4087) = X(3510)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b2c2(b + c - a) (a2 - bc)

X(4087) lies on these lines: {75, 982}, {312, 2321}, {314, 3706}, {350, 740}, {517, 668}, {646, 3693}, {1978, 3263}

### X(4088) = X(2254)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 - c2)(b2 + c2 - ab - ac)

X(4088) lies on these lines: {8, 2785}, {523, 661}, {676, 1639}, {918, 2254}, {1109, 2632}, {1282, 2786}, {1635, 2977}
X(4088) = anticomplement of X(4458)

### X(4089) = X(1023)com(INTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)2(b2 + c2 - a2 - bc)

X(4089) lies on these lines: {1, 7}, {11, 1111}, {495, 3665}, {527, 1023}, {1022, 2401}, {1086, 2087}, {3323, 3328}

### X(4090) = X(982)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(2a2 - ab - ac + bc)

X(4090) lies on these lines: {10, 12}, {43, 726}, {171, 3699}, {312, 519}, {537, 3752}, {3625, 3706}, {3681, 3741}

### X(4091) = X(663)com[INVERSE(n(INTANGENTS TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b - c)(b2 + c2 - a2)2

X(4091) lies on these lines: {101, 1262}, {110, 2727}, {239, 514}, {652, 905}, {654, 3669}, {663, 928}, {1734, 2812}

### X(4092) = X(662)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b2 - c2)2

X(4092) lies on these lines: {8, 645}, {10, 2652}, {115, 2643}, {125, 1109}, {281, 2175}, {542, 2607}, {1146, 3271}

X(4092) = X(7)-isoconjugate of X(1101)

### X(4093) = X(350)com(INCENTRAL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b + c)(b2 + c2)(a2 - bc)

X(4093) lies on these lines: {37, 42}, {38, 1964}, {350, 740}, {668, 718}, {688, 3005}, {1914, 2210}, {3009, 3726}

### X(4094) = X(335)com(INCENTRAL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)2(a2 - bc)2

X(4094) lies on the incentral inellipse and the inellipse centered at X(21254) and on these lines: {31, 643}, {37, 2054}, {42, 2107}, {238, 239}, {244, 1962}, {1100, 2309}, {2269, 2310}

X(4094) = antipode in the incentral inellipse of X(2643)
X(4094) = reflection of X(75) in X(21254)
X(4094) = reflection of X(2643) in X(37)
X(4094) = trilinear square of X(2238)

### X(4095) = X(39)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b + c - a)(a2 + bc)

X(4095) lies on these lines: {8, 2170}, {9, 341}, {10, 37}, {312, 3208}, {1018, 1089}, {1215, 2295}, {1334, 3701}

### X(4096) = X(3742)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(a2 - 2ab - 2ac + bc)

X(4096) lies on these lines: {2, 38}, {210, 740}, {341, 1089}, {519, 960}, {726, 3740}, {846, 3699}

### X(4097) = X(3739)com[INVERSE(n(INTOUCH TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(b + c - a)(2a2 - bc)

X(4097) lies on these lines: {10, 1001}, {42, 1449}, {55, 3686}, {71, 3174}, {100, 1014}, {3688, 3689}

### X(4098) = X(3731)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics    (b + c)(b + c - 7a) : :

Recalling that triangle centers are functions, at (a,b,c) = (6,9,13), the values of X(4098) and X(24150) are equal.

X(4098) lies on these lines: {1, 3161}, {10, 37}, {346, 1125}, {391, 519}, {551, 3247}, {1743, 3635}

### X(4099) = X(3691)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)2(2a2 - bc)

X(4099) lies on these lines: {192, 1930}, {740, 3294}, {1089, 1500}, {1224, 2345}, {1334, 2901}, {2171, 3671}

### X(4100) = X(1881)com[INVERSE(n(ORTHIC TRIANGLE))]

Trilinears       t(A,B,C) : t(B,C,A) : t(C,A,B), where t(A,B,C) = cos3A sin A    (Peter Moses, 1/13/2012)
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a5(b2 + c2 - a2)3

X(4100) lies on these lines: {19, 775}, {48, 255}, {63, 2148}, {163, 610}, {283, 2302}, {1820, 3708}

### X(4101) = X(1468)com[INVERSE(n(TANGENTIAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b + c + 3a)(b2 + c2 - a2)

X(4101) lies on these lines: {8, 226}, {10, 2650}, {69, 73}, {72, 306}, {391, 1449}, {3649, 3696}

### X(4102) = X(1051)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/(b + c + 2a)

X(4102) lies on these lines: {2, 594}, {8, 3058}, {190, 3578}, {257, 3175}, {333, 2321}, {519, 1126}

### X(4103) = X(1015)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)2/(b - c)

X(4103) lies on these lines: {10, 762}, {101, 3699}, {115, 594}, {514, 668}, {596, 1574}, {644, 3239}

### X(4104) = X(940)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b2 + c2 - a2 + 2ab + 2ac)

X(4104) lies on these lines: {2, 3751}, {10, 12}, {141, 3740}, {306, 756}, {984, 3687}, {3452, 3741}

### X(4105) = X(650)com[INVERSE(n(INTOUCH TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b - c)(b + c - a)3

X(4105) lies on these lines: {55, 652}, {109, 677}, {200, 3239}, {521, 2254}, {649, 926}, {650, 663}

### X(4106) = X(649)com(INTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)(a2 + ab + ac - 2bc)

X(4106) lies on these lines: {320, 350}, {514, 3700}, {522, 2526}, {650, 812}, {2786, 3776}, {3667, 3676}

X(4106) = complement of X(4380)
X(4106) = anticomplement of X(4394)

### X(4107) = X(649)com(1st BROCARD TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)(a4 - b2c2)

X(4107) lies on these lines: {239, 514}, {786, 2483}, {802, 3063}, {1027, 2665}, {1054, 2789}, {1919, 3261}

### X(4108) = X(351)com[INVERSE(n(1st BROCARD TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)(2a4 + b2c2)

X(4108) lies on these lines: {2, 512}, {98, 2770}, {351, 523}, {476, 1302}, {669, 804}, {2373, 2857}

X(4108) = X(351)-of-1st-anti-Brocard-triangle

### X(4109) = X(172)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b3 + c3 - a3 + abc)

X(4109) lies on these lines: {10, 213}, {37, 3178}, {226, 1231}, {1089, 1826}, {1330, 3509}, {3006, 3691}

### X(4110) = X(87)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c - a)(bc - ab - ac)

X(4110) lies on these lines: {8, 3056}, {9, 646}, {75, 141}, {312, 2321}, {341, 3704}, {668, 3729}

### X(4111) = X(86)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(b + c - a)(2bc + ab + ac)

X(4111) lies on these lines: {8, 314}, {210, 2321}, {2092, 3728}, {3271, 3686}, {3679, 3779}, {3704, 3717}

### X(4112) = X(31)com(1st BROCARD TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a5 + b3c2 + b2c3

X(4112) lies on these lines: {6, 714}, {31, 730}, {76, 1580}, {313, 560}, {1582, 3596}, {2210, 3765}

X(4112) = X(31)-of-1st-Brocard-triangle

### X(4113) = X(3720)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(2bc + 3ab + 3ac)

X(4113) lies on these lines: {8, 210}, {333, 3689}, {1215, 3626}, {3681, 3696}, {3686, 3693}

### X(4114) = X(3715)com(INTOUCH TRIANGLE)

Barycentrics  nbsp; f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (3b + 3c + 4a)/(b + c - a)

X(4114) lies on these lines: {2, 7}, {65, 3625}, {942, 3627}, {1319, 3671}, {2099, 3635}

### X(4115) = X(3125)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b + c + 2a)/(b - c)

X(4115) lies on these lines: {10, 115}, {37, 537}, {72, 2809}, {99, 101}, {213, 3159}

### X(4116) = X(3116)com(3rd BROCARD TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(ab3 + ac3 + b2c2)

X(4116) lies on these lines: {1, 76}, {32, 904}, {214, 995}, {766, 3116}, {1911, 2242}

### X(4117) = X(1978)com(INCENTRAL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a5(b2 - c2)2

X(4117) lies on these lines: {1, 799}, {31, 1927}, {42, 2107}, {561, 3223}, {669, 1977}

### X(4118) = X(1918)com(INCENTRAL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b4 + c4)

X(4118) lies on these lines: {1, 82}, {38, 1755}, {75, 1581}, {760, 1918}, {1959, 1964}

### X(4119) = X(1914)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b3 + c3 - 2abc)

X(4119) lies on these lines: {8, 41}, {10, 3290}, {594, 1575}, {2321, 3693}, {2348, 3686}

### X(4120) = X(1635)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics    (b + c - 2a)(b2 - c2) : :

X(4120) lies on these lines: {2, 2786}, {115, 125}, {523, 661}, {649, 3239}, {900, 1635}

X(4120) = isogonal conjugate of X(4591)
X(4120) = isotomic conjugate of X(4615)
X(4120) = tripolar centroid of X(10)
X(4120) = crossdifference of every pair of points on line X(58)X(106)
X(4120) = centroid of (degenerate) side-triangle of ABC and Gemini triangle 11

### X(4121) = X(1501)com[INVERSE(n(TANGENTIAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 + c2 - a2)(b4 + c4)

X(4121) lies on these lines: {51, 325}, {69, 184}, {125, 305}, {141, 1194}, {626, 3118}

### X(4122) = X(1491)com[INVERSE(n(TANGENTIAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 - c2)(b2 + c2 + bc)

X(4122) lies on these lines: {522, 659}, {523, 661}, {525, 2533}, {824, 1491}, {826, 1089}

### X(4123) = X(1397)com[INVERSE(n(INTOUCH TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(b4 + c4 - a4)

X(4123) lies on these lines: {1, 977}, {22, 1760}, {29, 33}, {345, 3100}, {643, 3719}

### X(4124) = X(1026)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b - c)2(a2 - bc)

X(4124) lies on these lines: {8, 210}, {11, 1146}, {239, 3573}, {242, 1428}, {514, 3675}

### X(4125) = X(995)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c)(2b + 2c - a)

X(4125) lies on these lines: {10, 321}, {312, 519}, {341, 3626}, {3625, 3702}, {3678, 3714}

### X(4126) = X(748)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b2 + c2 - 2ab - 2ac)

X(4126) lies on these lines: {8, 3058}, {10, 3782}, {200, 3712}, {210, 3687}, {345, 3711}

### X(4127) = X(3627)com[EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(3b2 + 3c2 - 3a2 - bc)

X(4127) lies on these lines: {10, 12}, {518, 3635}, {960, 3636}, {2802, 3632}

### X(4128) = X(668)com(INCENTRAL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b - c)(b2 - c2)(a2 + bc)

X(4128) lies on these lines: {1, 99}, {512, 1015}, {804, 3023}, {1356, 3122}, {2170, 2643}

### X(4129) = X(667)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 - c2)(a2 + ab + ac - bc)

X(4129) lies on these lines: {2, 1019}, {10, 512}, {514, 661}, {656, 3667}, {830, 3716}, {3814, 3836}

X(4129) = complement of X(1019)
X(4129) = pole wrt excircles-radical-circle of Brocard axis

### X(4130) = X(657)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b - c)(b + c - a)3

X(4130) lies on these lines: {9, 521}, {37, 2509}, {522, 650}, {644, 765}, {918, 3669}

X(4130) = isogonal conjugate of X(4617)
X(4130) = crossdifference of every pair of points on line X(56)X(269)

### X(4131) = X(650)com[INVERSE(n(TANGENTIAL TRIANGLE))]

Trilinears    (b - c) cot^2 A : :
Barycentrics    a(b - c)(b2 + c2 - a2)2 : :

X(4131) lies on these lines: {99, 2719}, {100, 677}, {320, 350}, {520, 3265}, {521, 4025}, {3676, 3738}

### X(4132) = X(513)com(INCENTRAL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 - c2)(a2 + ab + ac - bc)

X(4132) lies on these lines: {1, 3733}, {30, 511}, {37, 798}, {649, 3726}, {3050, 3063}

### X(4133) = X(3751)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) =(b + c)(b2 + c2 - 3a2 + 4bc)

X(4133) lies on these lines: {10, 37}, {306, 3120}, {519, 1992}

### X(4134) = X(381)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(2b2 + 2c2 - 2a2 + bc)

X(4134) lies on these lines: {8, 3583}, {10, 12}, {518, 551}, {519, 3681}, {960, 3244}

### X(4135) = X(43)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(ab + ac - 3bc)

X(4135) lies on these lines: {10, 321}, {312, 726}, {346, 3771}, {1215, 3175}, {3625, 3681}

### X(4136) = X(32)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b + c - a)(b2 + c2 - bc)

X(4136) lies on these lines: {8, 41}, {10, 37}, {626, 712}, {2887, 3721}, {3061, 3705}

### X(4137) = X(31)com(INCENTRAL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(b4 + c4 + ab3 + ac3)

X(4137) lies on these lines: {1, 2206}, {38, 1755}, {517, 2292}, {1962, 3744}, {2294, 3720}

### X(4138) = X(43)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(3b2 + 3c2 - a2 - 4bc)

X(4138) lies on these lines: {2, 1707}, {10, 12}, {306, 3120}, {516, 3771}

### X(4139) = X(3667)com(INCENTRAL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 - c2)(a2 + ab + ac - 2bc)

X(4139) lies on these lines: {30, 511}, {647, 1635}, {1643, 2489}, {3063, 3288}

X(4139) = isogonal conjugate of X(8690)
X(4139) = crossdifference of every pair of points on line X(6)X(404)

### X(4140) = X(3287)com(INCENTRAL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b2 - c2)(a2 + bc)

X(4140) lies on these lines: {37, 2395}, {522, 2321}, {523, 661}, {804, 2533}

### X(4141) = X(3120)com[INVERSE(n(4th BROCARD TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - 2a)(2b2 + 2c2 - a2)

X(4141) lies on these lines: {2, 726}, {519, 902}, {1647, 2325}, {2796, 3006}

### X(4142) = X(2530)com[INVERSE(n(HEXYL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)(b3 + c3 - a3 + abc)

X(4142) lies on these lines: {240, 522}, {514, 659}, {525, 3716}, {663, 2785}

### X(4143) = X(2485)com[INVERSE(n(TANGENTIAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 - c2)(b2 + c2 - a2)3

X(4143) lies on these lines: {69, 2419}, {99, 2764}, {520, 3265}, {525, 3267}

X(4143) = isotomic conjugate of X(6529)

### X(4144) = X(2243)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b3 + c3 - 2a3)

X(4144) lies on these lines: {10, 213}, {44, 3006}, {523, 661}, {752, 2243}

### X(4145) = X(900)com(INCENTRAL TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 - c2)(a2 + ab + ac - 3bc)

X(4145) lies on these lines: {30, 511}, {244, 2611}, {647, 2516}, {659, 3722}

### X(4146) = X(844)com(EXTOUCH TRIANGLE)

Trilinears    csc2A sin(A/2)    (Peter Moses, 1/13/2012)
Trilinears    |AIa| : |BIb| : |CIc|, where Ia = A-mixtilinear incenter
Trilinears    bc(sec A/2) = csc A sec A/2 : :

Barycentrics   sec(A/2) : sec(B/2) : sec(C/2)
Barycentrics   [bc/(b2 + c2 - a2 + 2bc)]1/2 : :

Let A'B'C' be the excentral triangle. Let Oa be the circle centered at A with radius |B'C'|, and define Ob and Oc cyclically. X(4146) is the trilinear pole of the Monge line of Oa, Ob, Oc. (Randy Hutson, June 27, 2018)

X(4146) lies on these lines: {7, 2091}, {85, 178}, {174, 556}, {188, 555}

X(4146) = isotomic conjugate of X(188)

X(4146) = cevapoint of X(174) and X(188)

### X(4147) = X(667)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)(b + c - a)(ab + ac - bc)

X(4147) lies on these lines: {8, 663}, {10, 514}, {522, 3717}, {1734, 3762}

### X(4148) = X(665)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)(b + c - a)(a2 - bc)

X(4148) lies on these lines: {812, 3766}, {966, 1769}, {1146, 2968}, {3686, 3738}

### X(4149) = X(604)com[INVERSE(n(INTOUCH TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(b3 + c3 - a3)

X(4149) lies on these lines: {1, 141}, {9, 2175}, {78, 1229}, {1631, 1759}

### X(4150) = X(560)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b4 + c4 - a4)

X(4150) lies on these lines: {10, 82}, {92, 264}, {315, 1760}, {344, 2478}

### X(4151) = X(514)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 - c2)(a2 - ab - ac - bc)

X(4151) lies on these lines: {10, 1577}, {30, 511}, {596, 876}, {693, 1734}

### X(4152) = X(88)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b + c - 2a)2

X(4152) lies on these lines: {8, 11}, {210, 3271}, {883, 3321}, {2325, 3689}

### X(4153) = X(32)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b3 + c3 - a3)

X(4153) lies on these lines: {10, 213}, {37, 3454}, {306, 1230}, {626, 742}

### X(4154) = X(3747)com(1st BROCARD TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(a2 - bc)(a4 - b2c2)

X(4154) lies on these lines: {238, 239}, {1045, 1178}, {1580, 1966}

### X(4155) = X(2786)com(INCENTRAL TRIANGLE)

Barycentrics    a(b + c)(b2 - c2)(a2 - bc) : :

X(4155) lies on these lines: {30, 511}, {351, 1635}, {2642, 2643}

X(4155) = ideal point of PU(i) for these i: 79, 143
X(4155) = crossdifference of every pair of points on line X(6)X(662)

### X(4156) = X(514)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b4 + c4 - 2a4)

X(4156) lies on these lines: {10, 82}, {523, 661}, {754, 2244}

### X(4157) = X(2244)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b4 + c4 - 2a4)

X(4157) lies on these lines: {8, 2175}, {522, 650}, {754, 2244}

### X(4158) = X(1612)com[INVERSE(n(TANGENTIAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b + c)2(b2 + c2 - a2)3

X(4158) lies on these lines: {72, 306}, {836, 3682}, {1018, 1490}

### X(4159) = X(1501)com(1st BROCARD TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a8 + b6c2 + b2c6

X(4159) lies on these lines: {184, 3734}, {384, 3117}, {736, 1501}

### X(4160) = X(1499)com(EXTOUCH TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b - c)(a2 + b2 + c2 + 3bc)

X(4160) lies on these lines: {1, 661}, {30, 511}, {1022, 1390}

X(4160) = isogonal conjugate of X(8691)
X(4160) = crossdifference of every pair of points on line X(6)X(896)

### X(4161) = X(1237)com(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(ab3 + ac3 + 2b2c2)

X(4161) lies on these lines: {1, 76}, {904, 1914}, {1201, 1279}

### X(4162) = X(905)com(INTANGENTS TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b - c)(b + c - a)(b + c - 3a)

X(4162) lies on these lines: {1, 3309}, {55, 667}, {650, 663}

### X(4163) = X(905)com(EXTOUCH TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)(b + c - a)3

X(4163) lies on these lines: {514, 2526}, {522, 3717}, {1016, 3699}

X(4163) = isotomic conjugate of X(4626)

### X(4164) = X(667)com(1st BROCARD TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b - c)(a4 - b2c2)

X(4164) lies on these lines: {36, 238}, {182, 2788}, {693, 1980}

### X(4165) = X(609)com(EXTOUCH TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b3 + c3 + abc)

X(4165) lies on these lines: {8, 41}, {1146, 3703}, {2170, 3705}

### X(4166) = X(510)com(EXTOUCH TRIANGLE))]

Trilinears       t(A,B,C) : t(B,C,A) : t(C,A,B), where t(A,B,C) = cot(A/2) sin1/2A    (Peter Moses, 1/13/2012)
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3/2(b + c - a)

X(4166) lies on these lines: {1, 2068}, {43, 2069}, {165, 364}

### X(4167) = X(172)com(EXTOUCH TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b3 + c3 + 2abc)

X(4167) lies on these lines: {8, 41}, {1146, 3704}, {2264, 3686}

### X(4168) = X(3721)com(EXTOUCH TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b3 + c3 + 2a3)

X(4168) lies on these lines: {8, 41}, {2264, 2321}

### X(4169) = X(2087)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b + c - 2a)/(b - c)

X(4169) lies on these lines: {10, 37}, {519, 2087}

### X(4170) = X(1019)com(INTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 - c2)(2a2 - bc)

X(4170) lies on these lines: {512, 1577}, {900, 2530}

### X(4171) = X(657)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 - c2)(b + c - a)2

X(4171) lies on these lines: {37, 656}, {523, 661}

X(4171) = isogonal conjugate of X(4637)
X(4171) = isotomic conjugate of X(4635)
X(4171) = crossdifference of every pair of points on line X(58)X(269)

### X(4172) = X(560)com(1st BROCARD TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a7 + b5c2 + b2c5

X(4172) lies on these lines: {76, 1933}, {560, 734}

### X(4173) = X(76)com[INVERSE(n(TANGENTIAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4(b4 + c4)(b2 + c2 - a2)

X(4173) lies on these lines: {3, 1808}, {39, 2387}

### X(4174) = X(1917)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b6 + c6 - a6)

X(4174) lies on this line: {10, 3407}

### X(4175) = X(1627)com[INVERSE(n(TANGENTIAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 + c2 - a2)(b2 + c2)2

X(4175) lies on this line: {125, 305}

### X(4176) = X(1611)com[INVERSE(n(TANGENTIAL TRIANGLE))]

Trilinears       t(A,B,C) : t(B,C,A) : t(C,A,B), where t(A,B,C) = cot3A csc A    (Peter Moses, 1/13/2012)
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 + c2 - a2)3

X(4176) lies on this line: {69, 305}

X(4176) = isotomic conjugate of X(6524)

### X(4177) = X(1501)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b5 + c5 - a5)

X(4177) lies on this line: {10, 2205}

### X(4178) = X(560)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b4 + c4)

X(4178) lies on this line: {8, 2175}

### X(4179) = X(367)com[INVERSE(n(INCENTRAL TRIANGLE))]

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a1/2(b + c)

X(4179) lies on this line: {365, 366}

### X(4180) = X(366)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a1/2(b1/2 + c1/2)(b + c - a)

X(4180) lies on this line: {1, 366}

### X(4181) = X(365)com(EXTOUCH TRIANGLE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b1/2 + c1/2)(b + c - a)

X(4181) lies on this line: {2, 366}

### X(4182) = X(364)com(EXTOUCH TRIANGLE)

Trilinears       t(A,B,C) : t(B,C,A) : t(C,A,B), where t(A,B,C) = csc2(A/2) sin1/2A    (Peter Moses, 1/13/2012)
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a1/2(b + c - a)

X(4182) lies on this line: {365, 366}

Points on the Euler Line: X(4183) - X(4250)

Many points are found as points of intersection of the Euler line and other central lines.

### X(4183) = INTERSECTION OF LINES X(2)X(3) AND X(9)X(33)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)2/[(b + c)(b2 + c2 - a2)]

As a point on the Euler line, X(4183) has Shinagawa coefficients (\$aSA\$F+abcF, \$aSA2\$-\$aSA\$(E+F) +\$a\$S2).

X(4183) lies on these lines: {1, 204}, {2, 3}, {9, 33}, {42, 1783}, {55, 281}, {81, 162}, {92, 1621}, {107, 972}, {108, 226}, {191, 1844}, {200, 1802}, {240, 846}, {278, 1001}, {282, 284}, {283, 2000}, {285, 1437}, {346, 1260}, {943, 1896}, {968, 1096}, {1430, 3720}, {1474, 2267}, {1490, 2360}, {1713, 1778}, {1728, 1780}, {1859, 3683}, {1864, 2194}, {2287, 2326}, {2906, 3193}

X(4183) = isogonal conjugate of X(1439)
X(4183) = polar conjugate of X(1446)

### X(4184) = INTERSECTION OF LINES X(2)X(3) AND X(35)X(42)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b2 + c2 - ab - ac + bc)/(b + c)
X(4184) = 3a2b2c2*X(2) + 4S2(bc + ca + ab)*X(3)

As a point on the Euler line, X(4184) has Shinagawa coefficients (2\$bc\$ + E, -2\$bc\$)

X(4184) lies on these lines: {2, 3}, {31, 3736}, {35, 42}, {36, 3720}, {55, 81}, {60, 1780}, {86, 1621}, {99, 310}, {100, 333}, {103, 110}, {228, 3219}, {283, 947}, {284, 672}, {476, 2688}, {573, 3060}, {846, 3724}, {917, 925}, {991, 1779}, {1014, 1617}, {1030, 2238}, {1043, 2975}, {1326, 2206}, {1333, 2276}, {1412, 2078}, {1444, 3433}, {1623, 1634}

### X(4185) = INTERSECTION OF LINES X(2)X(3) AND X(6)X(19)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 + ab + ac + 2bc)/(b2 + c2 - a2)

As a point on the Euler line, X(4185) has Shinagawa coefficients (\$a\$F, -\$a\$(E + F) - 2abc)

X(4185) lies on these lines: {1, 1824}, {2, 3}, {6, 19}, {9, 1868}, {12, 197}, {33, 1900}, {46, 1707}, {55, 1869}, {56, 225}, {57, 1426}, {275, 3597}, {278, 961}, {672, 2333}, {958, 1867}, {999, 1068}, {1155, 1878}, {1452, 1454}, {1474, 2278}, {1610, 3485}, {1753, 1902}, {1825, 2099}, {1836, 3556}, {1861, 1891}, {1968, 2204}, {3167, 3193}

### X(4186) = INTERSECTION OF LINES X(2)X(3) AND X(19)X(45)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 + ab + ac - 2bc)/(b2 + c2 - a2)

As a point on the Euler line, X(4186) has Shinagawa coefficients (\$a\$F, -\$a\$(E + F) + 2abc)

X(4186) lies on these lines: {1, 1828}, {2, 3}, {12, 1486}, {19, 45}, {33, 1829}, {34, 1319}, {55, 1842}, {56, 1877}, {100, 2899}, {108, 1398}, {208, 1876}, {242, 318}, {513, 1406}, {607, 2201}, {1068, 2969}, {1145, 1862}, {1210, 1473}, {1351, 3193}, {1452, 1887}, {1633, 1788}, {1837, 3556}, {1866, 2099}, {3214, 3217}

### X(4187) = INTERSECTION OF LINES X(2)X(3) AND X(10)X(11)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 - a2b2 - a2c2 + 2a2bc + 2ab2c + 2abc2 - 2b2c2

As a point on the Euler line, X(4187) has Shinagawa coefficients (abc\$a\$ - S2, -S2)

X(4187) lies on these lines: {1, 1329}, {2, 3}, {8, 496}, {10, 11}, {12, 1125}, {35, 3035}, {72, 1210}, {119, 1385}, {495, 3616}, {498, 1001}, {499, 958}, {908, 942}, {936, 3419}, {946, 3753}, {956, 2551}, {960, 1737}, {997, 1837}, {999, 3436}, {1376, 1479}, {1698, 1706}, {1834, 3216}, {3679, 3680}

### X(4188) = INTERSECTION OF LINES X(2)X(3) AND X(8)X(36)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2a3 - 2ab2 - 2ac2 + b2c + bc2 + abc)

As a point on the Euler line, X(4188) has Shinagawa coefficients (abc\$a\$ - 4S2, 4S2)

X(4188) lies on these lines: {1, 1392}, {2, 3}, {8, 36}, {35, 3616}, {55, 3622}, {56, 100}, {78, 3218}, {149, 3086}, {346, 2178}, {936, 3219}, {962, 2077}, {970, 2979}, {999, 3623}, {1055, 3501}, {1201, 3550}, {1376, 2975}, {1420, 3680}, {1468, 3240}, {1470, 3600}, {3306, 3601}

### X(4189) = INTERSECTION OF LINES X(2)X(3) AND X(8)X(35)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2a3 - 2ab2 - 2ac2 - b2c - bc2 - abc)

As a point on the Euler line, X(4189) has Shinagawa coefficients (abc\$a\$ + 4S2, -4S2)

X(4189) lies on these lines: {1, 89}, {2, 3}, {8, 35}, {36, 3616}, {55, 145}, {56, 1621}, {63, 3601}, {78, 3219}, {100, 958}, {956, 3621}, {966, 1030}, {970, 3060}, {1043, 1150}, {1125, 1770}, {1158, 3576}, {1727, 3612}, {3241, 3746}, {3295, 3623}

### X(4190) = INTERSECTION OF LINES X(2)X(3) AND X(8)X(46)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 - 3a4 + 2a2b2 + 2a2c2 - 2a2bc - 2ab2c - 2abc2 - 2b2c2

As a point on the Euler line, X(4190) has Shinagawa coefficients (abc\$a\$ - 2S2, 4S2)

X(4190) lies on these lines: {2, 3}, {7, 224}, {8, 46}, {56, 3434}, {65, 145}, {100, 388}, {390, 2646}, {391, 2245}, {528, 3304}, {950, 3306}, {997, 1770}, {1155, 3617}, {1320, 3296}, {1376, 3436}, {2550, 2975}, {3612, 3616}

### X(4191) = INTERSECTION OF LINES X(2)X(3) AND X(36)X(43)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a3b + a3c + a2bc - ab3 - ac3 - ab2c - abc2 - b3c - bc3 + 2b2c2)

As a point on the Euler line, X(4191) has Shinagawa coefficients (\$bc\$ - E, -\$bc\$)

X(4191) lies on these lines: {2, 3}, {6, 2350}, {36, 43}, {42, 56}, {51, 991}, {55, 750}, {57, 228}, {182, 1790}, {183, 310}, {198, 672}, {582, 1437}, {614, 2223}, {1155, 3185}, {1403, 3724}, {2178, 2276}, {2352, 3752}

### X(4192) = INTERSECTION OF LINES X(2)X(3) AND X(40)X(43)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a4b + a4c - a2b3 - a2c3 - 2ab2c2 - b4c - bc4 + b3c2 + b2c3)

As a point on the Euler line, X(4192) has Shinagawa coefficients (\$bcSBSC\$ + ES2, -\$bc\$S2)

X(4192) lies on these lines: {2, 3}, {40, 43}, {42, 517}, {165, 2108}, {182, 1754}, {228, 908}, {511, 1764}, {515, 3741}, {516, 2051}, {573, 2238}, {899, 3579}, {1385, 3720}, {1766, 2276}

### X(4193) = INTERSECTION OF LINES X(2)X(3) AND X(8)X(11)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b3 + c3 + ab2 + ac2 - abc - b2c - bc2)

As a point on the Euler line, X(4193) has Shinagawa coefficients (abc\$a\$ - 2S2, -S2).

X(4193) lies on these lines: {2, 3}, {8, 11}, {12, 1388}, {100, 1479}, {119, 944}, {145, 496}, {388, 1476}, {495, 3622}, {498, 1621}, {499, 2975}, {908, 1210}, {3086, 3436}, {3701, 3705}

### X(4194) = INTERSECTION OF LINES X(2)X(3) AND X(8)X(33)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(a2 + b2 + c2 + 2ab + 2ac)/(b2 + c2 - a2)

As a point on the Euler line, X(4194) has Shinagawa coefficients (\$a\$F, 2abc).

X(4194) lies on these lines: {2, 3}, {7, 208}, {8, 33}, {34, 3616}, {108, 3600}, {193, 3193}, {281, 318}, {391, 1172}, {393, 941}, {968, 1785}, {1788, 1887}, {1870, 3622}, {1875, 3485}

### X(4195) = INTERSECTION OF LINES X(2)X(3) AND X(8)X(31)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2a4 + a3b + a3c + a2bc + ab3 + ac3 + ab2c + abc2 + b3c + bc3 + 2b2c2

As a point on the Euler line, X(4195) has Shinagawa coefficients ((E + F)2 + \$bc\$(E + F) - S2, 2S2).

X(4195) lies on these lines: {1, 87}, {2, 3}, {6, 1043}, {8, 31}, {10, 3550}, {55, 1220}, {75, 1104}, {86, 1975}, {239, 1453}, {346, 2298}, {2322, 3172}, {3714, 3769}

### X(4196) = INTERSECTION OF LINES X(2)X(3) AND X(34)X(42)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 + c2 - a2 - 2ab - 2ac - 2bc)/(b2 + c2 - a2)

As a point on the Euler line, X(4196) has Shinagawa coefficients (F, -\$bc\$).

X(4196) lies on these lines: {2, 3}, {19, 672}, {33, 3720}, {34, 42}, {184, 3332}, {225, 1435}, {264, 310}, {278, 1002}, {393, 2350}, {1730, 1779}, {1836, 2182}, {1841, 2276}

### X(4197) = INTERSECTION OF LINES X(2)X(3) AND X(7)X(12)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 - a2b2 - a2c2 - 3a2bc - 3ab2c - 3abc2 - 2b2c2

As a point on the Euler line, X(4197) has Shinagawa coefficients (3abc\$a\$ + 2S2, 2S2).

X(4197) lies on these lines: {2, 3}, {7, 12}, {8, 3475}, {10, 3681}, {63, 1698}, {76, 3263}, {81, 1714}, {331, 1441}, {495, 3617}, {2346, 2550}, {2886, 3616}

X(4197) = homothetic center of 4th Euler triang