# BICENTRIC PAIRS OF POINTS

Definition of bicentric pair, P and U. Suppose f(a,b,c) : f(b,c,a) : f(c,a,b) is a point that satisfies requirement (1) in the definition of triangle center (in Glossary), but that |f(a,b,c)| is not equal to |f(a,c,b)|. Then

f(a,b,c) : f(b,c,a) : f(c,a,b)     and     f(a,c,b) : f(b,a,c) : f(c,b,a)

are a bicentric pair of points, or simply, a bicentric pair. Examples include the Brocard points, given in trilinear coordinates by

c/b : a/c : b/a     and     b/c : c/a : a/b.

In the list below, bicentric pairs are consecutively listed as P(1), P(2), ... . Along with each P(n), the second bicentric point U(n) is defined from the given f(a,b,c) as the point having first trilinear f(a,c,b). Barycentric coordinates for the two points are then

af(a,b,c) : bf(b,c,a) : cf(c,a,b)    and     af(a,c,b) : bf(b,a,c) : cf(c,b,a).

Following are operations which carry bicentric pairs P = p : q : r and U = u : v : w onto triangle centers. For two of these operations, it is necessary that p,q,r and u,v,w be represented by

f(a,b,c), f(b,c,a), f(c,a,b)    and    f(a,c,b), f(b,a,c), f(c,b,a).

Operation Trilinears
trilinear product pu : qv : rw
barycentric product apu: bqv: crw
bicentric sum p + u : q + v : r + w
bicentric difference p - u : q - v : r - w
crosssum qw + rv : ru + pw : pv + qu
crossdifference qw - rv : ru - pw : pv - qu
trilinear pole of line PU 1/(qw - rv) : 1/(ru - pw) : 1/(pv - qu)
ideal point of line PU p(bv+cw) - u(bq+cr) : q(cw+au) - v(cr+ap) : r(au+bv) - w(ap+bq)
midpoint kp + hu : kq + hv : kr + hw, where h = ap + bq + cr and k = au + bv + cw
cevapoint (pv + qu)(pw + ru) : (qw + rv)(qu + pv) : (ru + pw)(rv + qw)
crosspoint pu(rv + qw) : qv(pw + ru) : rw(qu + pv)
vertex conjugate a/[a2qrvw - pu(br + cq)(bw + cv)] : b/[b2rpwu - qv(cp + ar)(cu + aw)] : c/[c2pquv - rw(aq + bp)(av + bu)]

Following is an equivalent table using barycentric coordinates, P = p : q : r and U = u : v : w.

Operation Barycentrics
trilinear product bcpu : caqv : abrw
barycentric product pu: qv: rw
bicentric sum p + u : q + v : r + w
bicentric difference p - u : q - v : r - w
crosssum a2(qw + rv) : b2(ru + pw) : c2(pv + qu)
crossdifference a2(qw - rv) : b2(ru - pw) : c2(pv - qu)
trilinear pole of line PU 1/(qw - rv) : 1/(ru - pw) : 1/(pv - qu)
ideal point of line PU p(v+w) - u(q+r) : q(w+u) - v(r+p) : r(u+v) - w(p+q)
midpoint kp + hu : kq + hv : kr + hw, where h = p + q + r and k = u + v + w
cevapoint (pv + qu)(pw + ru) : (qw + rv)(qu + pv) : (ru + pw)(rv + qw)
crosspoint pu(rv + qw) : qv(pw + ru) : rw(qu + pv)
vertex conjugate a2/[a4qrvw - bcpu(b2r + c2q)(b2w + c2v)] : b2/[b4rpwu - qv(c2p + a2r)(c2u + a2w)] : c2/[c4pquv - rw(a2q + b2p)(a2v + b2u)]

It is easy to establish that a bicentric pair p : q : r and u : v : w lie on one and only one central line. Among triangle centers on this line are the bicentric sum and bicentric difference of the two points. The vertex conjugate of a pair PU is the P-vertex conjugate of U, which equals the U-vertex conjugate of V; see the Glossary of ETC for a definition of vertex conjugate. For a further discussion of the geometry associated with operations on bicentric pairs, see

C. Kimberling, Bicentric Pairs of Points and Related Triangle Centers, Forum Geometricorum 3 (2003) 35-47; Cubics Associated with Triangles of Equal Areas, Forum Geometricorum 1 (2001) 161-171; and "Enumerative triangle geometry, part 1: the primary system, S," Rocky Mountain Journal of Mathematics 32 (2002) 201-225. These articles are cited below as [Bicentric], [Areas], and [Enumerative], respectively.

Of the many bicentric pairs associated with a triangle center X = x : y : z (trilinears), the points XY and XZ defined by

XY = y : z : x    and    XZ = z : x : y

are the 1st and 2nd bicentrics of X, respectively. Following the method in [Areas], the 2nd equal-areas cubic is introduced here (Oct. 5, 2003) as the locus of a point X such that the areas of the cevian triangles of XY and XZ are equal. An equation for this cubic is

(bz + cx)(ay + cx)(ay + bz) = (bx + cy)(az + cy)(az + bx),

that is, the cubic K155. (The 1st equal-areas cubic, K021, is the locus of X for which the cevian triangles of X and its isogonal conjugate have equal areas.)

In barycentric coordinates, if X = x : y : z, then the points Xy and Xz defined on page 85 of

John Casey, A Treatise on the Analytic Geometry of the Point, Line, Circle, and Conic Sections, 2nd edition, Hodges, Figgis, Dublin, 1893

by Xy = y : z : x and Xz = z : x : y are a bicentric pair (if X is not the centroid) here named the 1st and 2nd isobarycs of X, respectively. Casey calls the set {X, Xy, Xz} an isobaryc group of points and notes that the triangle formed by an isobaryc group is triply perspective to the reference triangle. If X = x : y : z (trilinears), then trilinears for the isobarycs are given by

Xy = by/a : cz/b : ax/c    and    Xz = cz/a : ax/b : by/c.

As the list of bicentric pairs continues to grow, it is convenient to borrow names from another field, one whose objects, like those of triangle geometry, please and proliferate. In keeping with the spirit of Hyacinthos message 7999, captioned "Another stellar (or flowered) transformation", names of flowers are selected for certain bicentric pairs, such as Acacia points for PU(43) . A special case is the flower name Hyacinth, which, according to The Language of Flowers - alphabetical by flower name, means the language of flowers. The symmedian point, X(6), also known as the Lemoine point in honor of Emile Michel Hyacinthe Lemoine, has bicentrics P(6) and U(6), which may be called the hyacinth points.

#### P(1) = 1st BROCARD POINT

Trilinears    c/b : a/c : b/a
Barycentrics    1/b2 : 1/c2 : 1/a2

The 1st and 2nd Brocard points appear prominently in the well-known books by Johnson, Altshiller-Court, Gallatly, and Honsberger. In the Historical and Bibliographical Notes near the end of Altshiller-Court, Henri Brocard's contributions regarding the points that now bear his name are traced back to 1875, although the two points had been encountered by previous writers.

If you have The Geometer's Sketchpad, you can view 1st BROCARD POINT. For an online account, see Section 6.4 in

Yiu also notes, in Section 8.4, a generalization of the two Brocard points, called Brocardians: if X = x : y : z (trilinears) is a triangle center, then the Brocardians of X are the points

b/z : c/x : a/y and c/y : a/z : b/x.

(If X is represented in barycentric coordinates as x : y : z, then the Brocardians of X are 1/z : 1/x : 1/y and 1/y : 1/z : 1/x. In Section 8.4, Yiu shows a construction for these points.)

Definition: The 2nd Brocard circle is the circle having center X(3) and radius eR, where e = (1 - 4 sin2ω)1/2 and R = circumradius. The 2nd Brocard circle and the Brocard circle meet in the Brocard points. [The 2nd Brocard circle also passes through X(1670) and X(1671); see notes in ETC before X(1662) and before X(2446).] (Peter J. C. Moses, 10/22/03)

PU(1) are the 1st and 2nd bicentric quotients (defined at P(113)) of the centroid of ABC; hence, they are the 2nd and 1st bicentric quotients of the symmedian point.

The center of the circumconic {{A,B,C,PU(1)}} is X(385).

P(1) and U(1) lie on the central line X(39)X(512), which is parallel to PU(2).
P(1) lies on these non-central lines: X(3)P(2), X(6)U(39), X(32)U(2), X(880)U(11), X(1083)P(26), X(1316)P(157), X(5091)P(8), P(9)P(49), U(27)P(47), P(91)U(133), P(148)P(159), U(155)U(157) (Randy Hutson, April 9, 2016)

P(1) lies on these non-central lines: X(1)U(146), X(384)P(156), X(3118)U(156), X(10684)P(155), X(14382)U(38), X(14947)U(114) P(11)P(167) P(37)P(45) P(38)U(40) U(44)P(46) U(89)U(109) P(91)U(133) U(91)U(140) P(148)P(159). (Randy Hutson, July 31 2018)

P(1) lies on these non-central lines: X(76)U(178), X(194)P(178), P(43)P(186), P(154)U(181). (Randy Hutson, August 29 2018)

P(1) = 1st bicentric of P(10) = 2nd bicentric of U(9)
U(1) = 1st bicentric of P(9) = 2nd bicentric of U(10)
P(1) = P(1)-of-pedal-triangle-of-P(1)
P(1) = P(1)-of-antipedal-triangle-of-P(1)
P(1) = P(1)-of-reflection-triangle-of-P(1)
P(1) = X(3)-of-reflection-triangle-of-U(1)
P(1) = trilinear pole of line X(385)U(145)
P(1) = crossdifference of every pair of points on line X(385)P(145)
P(1) = 1st isobaryc of X(76)
P(1) = 2nd isobaryc of U(1)
P(1) = U(1)-of-circumcevian-triangle-of-P(1)

#### P(2) = 1st BELTRAMI POINT

Trilinears   a(b2 - a2) : :
Barycentrics   a2(b2 - a2) : :

P(2) and U(2) are the circumcircle-inverses of P(1) and U(1), respectively.
The center of the circumconic {{A,B,C,PU(2)}} is X(110).

Frank Morley and F. V. [Frank Vigor] Morley, Inversive Geometry, (originally published in 1933), Chelsea reprint, 1954, page 78, write, "They might be called Beltrami points and their join the Beltrami line, since they occur in Beltrami's memoir, Mem. della Accad. di Bologna, ser. 2, v. 9 (1870), where the theory of the triangle was first adequately discussed."

P(2) and U(2) lie on the central line X(187)X(237), which is parallel to PU(1).
P(2) lies on these non-central lines: X(3)P(1), X(32)U(1), X(691)P(105), X(805)P(91), X(3053)U(39)

P(2) = trilinear pole of line X(110)P(62)

#### P(3) = 1st YFF POINT

Trilinears   bc[(c - u)/(b - u)]1/3 : : , where u is the (only) real root of the cubic polynomial t3 + (t - a)(t - b)(t - c)
Barycentrics   [(c - u)/(b - u)]1/3 : :

The bicentric pair P(3), U(3) are introduced and described in

Peter Yff, "An Analog of the Brocard Points," American Mathematical Monthly 70 (1963) 495-501;

The line PU(3) is perpendicular to the line X(1)X(3) and parallel to the line X(1)X(513). The lines PU(3), PU(6), PU(31), and PU(33) are parallel. (Randy Hutson, September 10, 2012)

If you have The Geometer's Sketchpad, you can view Yff Points.

#### P(4)  1st GRINBERG INTERSECTION

Trilinears   [cot B + cot C - 2 cot A + (tan B - tan C) L1/2] sec A : : , where L = - cot A cot B cot C (cot A + cot B + cot C)
Barycentrics    [cot B + cot C - 2 cot A + (tan B - tan C) L1/2] tan A : :
Barycentrics    SA^3 - (4*SB*SC + SW^2)*SA + (SA + SW)*S^2 - (SB - SC)*sqrt(-SA*SB*SC*SW) : :
(César Lozada, December 08, 2021)

The circumcircle and nine-point circle of triangle ABC meet in real points if and only if ABC is not acute. (Darij Grinberg, Hyacinthos 6836, March 29, 2003)

P(4) and U(4) lie on the central line X(230)X(231).

The lines PU(4), PU(5), PU(11), and PU(45) are parallel. Also, the points PU(4) lie on the orthocentroidal circle and on the orthic axis. (Randy Hutson, September 10, 2012)

P(4) and U(4) are a pair of X(2)-Ceva conjugates. (Randy Hutson, November 22, 2014)

#### P(5) = 1st EHRMANN PIVOT

Trilinears    sin(B - C - π/3) : :
Barycentrics    (sin A)sin(B - C - π/3) : :

P(5) has the following remarkable property: the rotation through angle 2π/3 carries triangle ABC onto a triangle circumscribing and congruent to triangle ABC, and the rotation through - 2π/3 carries ABC onto a triangle inscribed in and congruent to ABC. The same is true using U(5) as center of rotation.

The Euler line bisects segment P(5)U(5) in the nine-point center, X(5). (See the table following P(6)).

Contributed by Jean-Pierre Ehrmann, April 22, 2003.

If you have The Geometer's Sketchpad, you can view EHRMANN PIVOTS.

P(5) and U(5) lie on the central line X(5)X(523).

The points PU(5) lie on the orthocentroidal circle. (Randy Hutson, September 10, 2012)

P(5) lies on these non-central lines: X(623)P(11), X(625)U(11), X(18581)P(45), X(18582)U(45). (Randy Hutson, July 31, 2018) 2012)

P(5) lies on these non-central lines: X(2)P(175), X(3)U(174), X(4)U(175), X(381)P(174). (Randy Hutson, August 29, 2018) 2012)

P(5) = 1st Ehrmann pivot of 1st Ehrmann circumscribing triangle
P(5) = 1st Ehrmann pivot of 1st Ehrmann inscribed triangle

#### P(6) = 1st BICENTRIC OF X(6)

Trilinears    b : c : a
Barycentrics    1/c : 1/a : 1/b

These are the Brocardians (as mentioned at P(1)) of the incenter; they are also known as the Jerabek points and the Hyacinth points.

P(6) and U(6) lie on the central line X(37)X(513).
P(6) lies on these non-central lines: X(1)P(31), X(6)U(31), X(2607)P(15), X(3570)U(8), X(3571)P(8)

The line PU(6) is perpendicular to the line X(1)X(3). (Randy Hutson, September 10, 2012)

P(6) = perspector of the 1st bicentric of the circumcircle

#### P(7) = 1st EVANS-YFF POINT

Trilinears    bc(b4 - b2c2 + 2c2a2 - 3a2b2) : :
Barycentrics    (b4 - b2c2 + 2c2a2 - 3a2b2) : :

Let [W,r] denote the circle of radius r and center W. The radical axes of the circles

[A,|AB|],    [B,|BC|],   [C,|CA|]

meet in P(7). Likewise, the radical axes of the circles

[A,|AC|],    [B,|BA|],   [C,|CB|]

meet in U(7). The pair P(7), U(7), discovered in 1998 by Lawrence Evans and Peter Yff, are described in TCCT, page 60.

P(7) and U(7) lie on the central line X(5)X(1499).

The column headings in the following table abbreviate the operations defined above. For example, in row 1, the appearance of 384 means that center X(384) in ETC is the crosssum of the bicentric pairs P(1),U(1).

Index tri prd bary prd bic sum bic diff crsum crdiff tri pole ideal pt midpt cevapt crspt vertcon
1 1 6 39 512 384 385 694 512 39 . . .
2 163 1576 187 512 148 2 6 512 187 . . .
3 75 2 . . . . . 513 . . . .
4 19 25 468 523 6 3 4 523 468 . . .
5 . . 5 523 567 50 94 523 5 . . .
6 2 1 37 513 171 238 291 513 37 . . .
7 . . 5 1499 . . . 1499 5 . . .

#### P(8) = 1st BICENTRIC OF X(2)

Trilinears    1/b : 1/c : 1/a
Barycentrics    a/b : b/c : c/a

P(8) and U(8) lie on the central line X(42)X(649).
P(8) lies on these non-central lines: X(1)P(84), X(3570)U(6), X(3571)P(6), X(5091)P(1), P(15)P(16) (the 1st bicentric of the Euler line)

Suppose that L = VW is a central line. Let V' = 1st bicentric of V and W' = 1st bicentric of W. Then V'W' is a line introduced here as the 1st bicentric of L, and likewise for the 2nd bicentric of L. Let LL = line at infinity = X(30)X(511). The 1st bicentric of LL is the line P(33)P(49) , which passes through X(238); then 2nd bicentric of LL is the line U(33)U(49), which also passes through X(238). P(8) = trilinear pole of 1st bicentric of LL, and U(8) = trilinear pole of 2nd bicentric of the LL. (Randy Hutson, December 26, 2015)

#### P(9) = TRILINEAR PRODUCT X(6)*P(8)

Trilinears    a/b : b/c : c/a
Barycentrics    a3c : b3a : c3b

P(9) and U(9) lie on the central line X(213)X(667).
P(9) lies on these non-central lines: P(1)P(49), , P(10)U(90).

P(9) = 2nd bicentric of U(1) = 1st bicentric of U(10)
U(9) = 1st bicentric of P(1) = 2nd bicentric of P(10)
P(9) = isogonal conjugate of P(10)

#### P(10) = TRILINEAR PRODUCT X(2)*P(6)

Trilinears    b/a : c/b : a/c
Barycentrics    b : c : a

P(10) and U(10) are the isobarycs of the incenter. Thus, by Casey's perspectivity theorem, triangle X(1)P(10)U(10) is triply perspective to triangle ABC.

P(10) and U(10) lie on the central line X(10)X(514).

P(10) = 2nd bicentric of P(1) = 1st bicentric of U(9)
U(10) = 1st bicentric of U(1) = 2nd bicentric of P(9)
P(10) = X(4)-of-1st-Montesdeoca-bisector-triangle
U(10) = X(4)-of-2nd-Montesdeoca-bisector-triangle
P(10) = isogonal conjugate of P(9)

#### P(11) = ISOTOMIC CONJUGATE OF 1st BROCARD POINT

Trilinears    b2/a : c2/b : a2/c
Barycentrics    b2 : c2 : a2

P(11) and U(11) lie on the central line X(141)X(523).
P(11) lies on these non-central lines: X(2)P(45), X(6)U(179), X(69)U(45), X(623)P(5), X(625)U(5), X(880)U(1), X(3620)U(132), X(7774)U(43), X(7999)U(61), X(11174)P(43) P(1)P(167).

P(11) = 1st isobaryc of X(6)
P(11) = perspector of 1st isobaryc of circumcircle
P(11) = complement of P(45)

The perspector of conic {{A,B,C,PU(11)}} is X(3978). (Randy Hutson, April 9, 2016)

#### P(12) = TRILINEAR PRODUCT X(6)*P(9)

Trilinears    a2/b : b2/c : c2/a
Barycentrics    a3/b : b3/c : c3/a

P(12) and U(12) lie on the central line X(1918)X(1919).
The perspector of the circumconic {{A,B,C,PU(12)}} is X(2210).

#### P(13) = TRILINEAR SQUARE OF P(9)

Trilinears    a2/b2 : b2/c2 : c2/a2
Barycentrics    a3/b2 : b3/c2 : c3/a2

P(13) and U(13) lie on the central line X(1923)X(1924).
The perspector of the circumconic {{A,B,C,PU(13)}} is X(1933).
P(13) = isogonal conjugate of P(14)

#### P(14) = TRILINEAR SQUARE OF P(10)

Trilinears    b2/a2 : c2/b2 : a2/c2
Barycentrics    b2/a : c2/b : a2/c

P(14) and U(14) lie on the central line X(1577)X(1930).
The perspector of the circumconic {{A,B,C,PU(14)}} is X(1926).
P(14) = isogonal conjugate of P(13)

Index tri prd bary prd bic sum bic diff crsum crdiff tri pole ideal pt midpt cevapt crspt vertcon
8 6 31 42 649 894 239 292 788 1908 . . .
9 32 560 213 667 1909 350 1911 1912 1913 . . .
10 76 75 10 514 172 1914 335 514 10 257 1909 .
11 561 76 141 523 1915 1691 1916 523 141 . . .
12 1501 1917 1918 1919 1920 1921 1922 . . . . .
13 1917 . 1923 1924 1925 1926 1927 . . . . .
14 1928 1502 1930 1577 . 1933 1934 9237 . . . .

#### P(15) = 1st BICENTRIC OF X(3)

Trilinears    cos B : cos C : cos A
Barycentrics    sin A cos B : sin B cos C : sin C cos A

PU(15) are the polar conjugates of PU(20).

P(15) and U(15) lie on the central line X(65)X(650).
P(15) lies on these non-central lines: X(2607)P(6) (the 1st bicentric of the Brocard axis), and P(8)P(15) (the 1st bicentric of the Euler line)

The perspector of the circumconic {{A,B,C,PU(15)}} is X(243).

P(15) = isogonal conjugate of P(16)

#### P(16) = 1st BICENTRIC OF X(4)

Trilinears    sec B : sec C : sec A
Barycentrics    sin a sec B : sin B sec C : sin C sec A

P(16) and U(16) lie on the central line X(73)X(652).
P(16) lies on non-central line P(8)P(15) (the 1st bicentric of the Euler line).

The points PU(6) lie on the circumconic {A,B,C,X(21),X(651)}. (Randy Hutson, September 10, 2012)

P(16) = isogonal conjugate of P(15)

#### P(17) = TRILINEAR PRODUCT P(15)*U(16)

Trilinears    cos B sec C : cos C sec A : cos A sec B
Barycentrics    sin A cos B sec C : sin B cos C sec A : sion C cos A sec B

PU(17) are the 1st and 2nd bicentric quotients (defined at P(113)) of the circumcenter of ABC; hence, they are the 2nd and 1st bicentric quotients of the orthocenter).

P(17) and U(17) lie on the central line X(185)X(647).
P(17) lies on these non-central lines: X(4)U(157), X(389)P(157).

The perspector of the circumhyperbola {{A,B,C,PU(17)}} is X(450).

P(17) = isogonal conjugate of U(17)

#### P(18) = TRILINEAR PRODUCT X(6)*P(15)

Trilinears    cos B sec C : cos C sec A : cos A sec B
Barycentrics    sin A cos B sec C : sin B cos C sec A : sin C cos A sec B

P(18) and U(18) lie on the central line X(663)X(1400).
The perspector of the circumconic {{A,B,C,PU(18)}} is X(2202).

#### P(19) = TRILINEAR PRODUCT X(6)*P(16)

Trilinears    a sec B : b sec C : c sec A
Barycentrics    a2sec B : b2sec C : c2sec A

PU(19) lies on the line X(1409)X(1946). (Randy Hutson, November 22, 2014)
The perspector of the circumconic {{A,B,C,PU(19)}} is X(1951).

P(19) = isogonal conjugate of P(20)

#### P(20) = TRILINEAR PRODUCT X(2)*P(15)

Trilinears    cos B csc A : cos C csc B : cos A csc C
Barycentrics    cos B : cos C : cos A

PU(20) are the polar conjugates of PU(15).

P(20) and U(20) lie on the central line X(226)X(522).
The perspector of the circumconic {{A,B,C,PU(20)}} is X(1948).

P(20) = isogonal conjugate of P(19)

#### P(21) = 1st BICENTRIC OF X(48)

Trilinears    sin 2B : sin 2C : sin 2A
Barycentrics    sin2A cos B : sin2B cos C : sin2C cos A

P(21) and U(21) lie on the central line X(656)X(1953).

#### P(22) = 1st BICENTRIC OF X(19)

Trilinears    tan B : tan C : tan A
Barycentrics    sin A tan B : sin B tan C : sin C tan A

P(22) and U(22) lie on the central line X(48)X(656). The isogonal conjugates of PU(22) are PU(23).

#### P(23) = 1st BICENTRIC OF X(63)

Trilinears    cot B : cot C : cot A
Barycentrics    sin A cot B : sin B cot C : sin C cot A

P(23) and U(23) lie on the central line X(31)X(661). The isogonal conjugates of PU(23) are PU(22).
The perspector of the circumhyperbola {{A,B,C,PU(12)}} is X(240).

Index tri prd bary prd bic sum bic diff crsum crdiff tri pole ideal pt midpt cevapt crspt vertcon
15 4 19 65 650 1935 1936 1937 1938 1939 . . .
16 3 48 73 652 1940 243 296 . . . . .
17 1 6 185 647 1941 450 1942 . . . . .
18 25 1973 1400 663 1943 1944 1945 . . . . .
19 184 . 1409 1946 1947 1948 1949 . . . . .
20 264 92 226 522 1950 1951 1952 522 226 . . .
21 92 4 1953 656 1954 1955 1956 . . . . .
22 63 3 48 656 1957 240 293 . . . . .
23 19 25 31 661 1958 1959 1910 . . . . .

#### P(24) = 1st VEGA BICENTRIC OF X(6)

Trilinears    1 - b/a : 1 - c/b : 1 - a/c
Barycentrics    a - b : b - c : c - a

Suppose X = x : y : z is a triangle center other than the incenter. The Vega transform of X is defined in ETC (at X(1981)) as the point

(y - z)/x : (z - x)/y : (x - y)/z.

The Vega bicentrics of X are here introduced by the trilinears:

1st Vega bicentric:    (x - y)/x : (y - z)/y : (z - x)/z

2nd Vega bicentric:    (x - z)/x : (y - x)/y : (z - y)/z.

The three Vega points are collinear; indeed, their line has trilinear coefficients x,y,z and is the trilinear polar of the isogonal conjugate of X.

P(24) and U(24) lie on the line at infinity, X(30)X(511).
P(24) lies on non-central line X(239)P(58).

P(24) = crossdifference of every pair of points on line P(8)U(48)
U(24) = crossdifference of every pair of points on line U(8)P(48)
P(24) = isogonal conjugate of P(25)

#### P(25) = ISOGONAL CONJUGATE OF P(24)

Trilinears    a/(a - b) : b/(b - c) : c/(c - a)
Barycentrics    a2/(a - b) : b2/(b - c) : c2/(c - a)

P(25) and U(25) lie on the circumcircle and on the line X(1015)X(1960).
P(25) lies on non-central line X(1)P(27).

P(25) = trilinear pole of line X(6)P(8)U(48)
U(25) = trilinear pole of line X(6)U(8)P(48)

#### P(26) = 1st VEGA BICENTRIC OF X(2)

Trilinears    1 - a/b : 1 - b/c : 1 - c/a
Barycentrics    a(1 - a/b) : b(1 - b/c) : c(1 - c/a)

P(26) and U(26) lie on the central line X(187)X(237), which is also P(2)U(2).
P(26) lies on these non-central lines: X(238)P(33), X(1083)P(1)

P(26) = isogonal conjugate of P(27)

#### P(27) = ISOGONAL CONJUGATE OF P(26)

Trilinears    b/(b - a) : c/(c - b) : a/(a - c)
Barycentrics    ab/(b - a) : bc/(c - b) : ca/(a - c)

P(27) and U(27) lie on the central line X(244)X(665).

PU(27) lie on the Steiner circumellipse. (Randy Hutson, November 22, 2014)
P(27) lies on these non-central lines: X(1)P(25), U(1)U(47).

P(27) = trilinear pole of line X(2)P(6)

#### P(28) = 1st VEGA BICENTRIC OF X(513)

Trilinears    (a + b - 2c)/(b - c) : (b + c - 2a)/(c - a) : (c + a - 2b)/(a - b)
Barycentrics    a(a + b - 2c)/(b - c) : b(b + c - 2a)/(c - a) : c(c + a - 2b)/(a - b)

P(28) and U(28) lie on the central line X(1)X(6).
The perspector of the circumconic {{A,B,C,PU(28)}} is X(3257).

#### P(29) = 1st VEGA BICENTRIC OF X(523)

Trilinears    1 + sin(A - C)/sin(B - C) : :
Barycentrics    (sin A)(1 + sin(A - C)/sin(B - C)) : :

P(29) and U(29) lie on the Brocard axis, X(3)X(6).

#### P(30) = 1st VEGA BICENTRIC OF X(647)

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

P(30) and U(30) lie on the Euler line, X(2)X(3).

Index tri prd bary prd bic sum bic diff crsum crdiff tri pole ideal pt midpt cevapt crspt vertcon
24 668 190 519 514 . 6 2 . . . . .
25 667 1919 1960 1015 . 190 649 . . . . .
26 101 692 3230 667 . 2 6 512 . . . .
27 514 513 891 1015 . 100 513 891 1015 . . .
28 5376 . 1 1023 . 513 100 518 . . . .
29 . . . 1983 . 523 110 . . . . .
30 . . 1982 1981 . 647 648 30 . . . .

#### P(31) = 1st BICENTRIC OF X(37)

Trilinears    a + c : b + a : c + b
Barycentrics    a(a + c) : b(b + a) : c(c + b)

P(31) and U(31) lie on the central line X(513)X(1100), which is perpendicular to line X(1)X(3).
P(31) lies on these non-central lines: X(1)P(6), X(6)U(6), X(9)U(52), X(37)U(33), X(894)P(41), X(1386)U(96), X(1449)U(46), X(4360)U(41), X(16666)U(50), X(16777)P(52), X(16884)P(53) P(8)P(36) P(22)P(94) U(45)P(47). (Randy Hutson, July 31, 2018)

The perspector of the circumconic {{A,B,C,PU(31)}} is X(1931).
P(31) = isogonal conjugate of P(32)

#### P(32) = 1st BICENTRIC OF X(81)

Trilinears    1/(a + c) : 1/(b + a) : 1/(c + b)
Barycentrics    a/(a + c) : b/(b + a) : c/(c + b)

P(32) and U(32) lie on the central line X(661)X(1962).
P(32) lies on non-central line P(34)P(84).

The perspector of the circumconic {{A,B,C,PU(32)}} is X(1757).
P(32) = isogonal conjugate of P(31)

#### P(33) = 1st BICENTRIC OF X(513)

Trilinears    a - c : b - a : c - b
Barycentrics    a(a - c) : b(b - a) : c(c - b)

P(33) and U(33) lie on the central line X(44)X(513) (the antiorthic axis).
P(33) lies on these non-central lines: X(1)U(52), X(6)U(6), X(9)P(6), X(37)U(31), X(45)P(52), X(190)U(41), X(1016)U(24), X(238)P(26), X(239)P(41), X(518)U(96), X(1743)P(46), X(3257)U(34), X(16669)P(54), X(16885)P(53). (Randy Hutson, July 31, 2018)

The line PU(33) is perpendicular to the line X(1)X(3).
P(33) = isogonal conjugate of P(34)

#### P(34) = 1st BICENTRIC OF X(100)

Trilinears    1/(a - c) : 1/(b - a) : 1/(c - b)
Barycentrics    a/(a - c) : b/(b - a) : c/(c - b)

P(34) and U(34) lie on the central line X(244)X(665), which is also P(27)U(27).
P(34) lies on non-central line P(32)P(84).

The points PU(34) lie on the circumconic having center X(9) and passing through X(100), X(658), X(662), X(799), X(1821), X(2580), and X(2581).
P(34) = isogonal conjugate of P(33)

#### P(35) = 1st BICENTRIC OF X(31)

Trilinears    b2 : c2 : a2
Barycentrics    ab2 : bc2 : ca2

P(35) and U(35) lie on the central line X(38)X(661).
The perspector of the circumconic {{A,B,C,PU(35)}} is X(1966).

P(35) = 1st isobaryc quotient of X(1)
P(35) = 2nd isobaryc quotient of X(75)
P(35) = isogonal conjugate of P(36)

#### P(36) = 1st BICENTRIC OF X(75)

Trilinears    1/b2 : 1/b2 1/b2
Barycentrics    a/b2 : b/b2 c/b2

P(36) and U(36) lie on the central line X(513)X(1834).
P(36) = isogonal conjugate of P(35)

Index tri prd bary prd bic sum bic diff crsum crdiff tri pole ideal pt midpt cevapt crspt vertcon
31 81 58 1100 513 1961 1757 1929 513 1100 . . .
32 37 42 1962 661 1963 1931 . . . . . .
33 100 101 44 513 1054 1 1 513 44 . . .
34 513 649 1635 244 6163 100 513 891 . . . .
35 75 2 38 661 1582 1580 1581 . . . . .
36 31 32 1964 798 1965 1966 1967 . . . . .

#### P(37) = 1st VERTEX PRODUCT OF HEX(31-36)

Trilinears    b/(a cos B) : c/(b cos B) : a/(c cos B)
Barycentrics    tan B : tan C : tan A

In [Enumerative], objects 31-36 comprise a hex generated from {A,B,C}. These six objects are algebraic entities that can be interpreted as coordinates of points, or, dually, as coefficients of lines. In the former case, objects 31,32,33 are vertices of a triangle and objects 34,35,36 are vertices of a second triangle. The two comprise a bicentric pair of triangles. The trilinear product of the first three vertices is point P(37), and the product of the other three, point U(37). Vertex products are discussed in section 11 of [Bicentric]; where, in row 1 of Table 4, the hex {31,32,33,34,35,36}, the vertex products P(37) and U(37) and the product P(37)*U(37) of all six objects (a triangle center), appear in abbreviated form.

PU(37) are the pedal antipodal perspectors of PU(1); see X(5485). (Randy Hutson, January 5, 2014)

P(37) and U(37) lie on the central line X(3)X(525).
P(37) lies on these non-central lines: X(2)P(38) (the 1st isobaryc of the Euler line), X(4)U(38), X(20)U(131), P(109)U(133), X(522)U(44), X(877)U(39), X(1370)U(170) P(1)P(45) P(109)U(133). (Randy Hutson, July 31, 2018)

P(37) lies on these non-central lines: U(17)P(178) P(168)P(172). (Randy Hutson, August 29, 2018).

P(37) = 1st isobaryc of X(4)
P(37) = isotomic conjugate of P(45)

The points PU(37) are isogonal conjugates of the points PU(39).
The perspector of the circumhyperbola {{A,B,C,PU(37)}} is X(325).

#### P(38) = 1st VERTEX PRODUCT OF HEX(43-48)

Trilinears    (b cos B)/a : (c cos C)/b : (a cos A)/c
Barycentrics    sin 2B : sin 2C : sin 2A

See the description at P(7) and row 3 of Table 4 in [Bicentric].

P(38) and U(38) lie on the central line X(5)X(525).
P(38) lies on these non-central lines: X(2)P(37) (the 1st isobaryc of the Euler line), X(4)U(37), X(512)P(7), X(3091)U(131), X(3566)U(7), X(5133)U(170), X(14382)U(1), P(1)U(40),P(135)P(136). (Randy Hutson, July 31, 2018)

P(38) lies on these non-central lines: X(381)U(176), P(168)P(171). (Randy Hutson, August 29, 2018).

P(38) = 1st isobaryc of X(3)
P(38) = polar conjugate of P(157)

#### P(39) = 1st VERTEX PRODUCT OF HEX(110-115)

Trilinears    a cot B : b cot C : c cot A
Barycentrics    a2 cot B : b2 cot C : c2 cot A

See the description at P(7) and row 11 of Table 4 in [Bicentric].

P(39) and U(39) lie on the central line X(32)X(512).
P(39) lies on these non-central lines: X(3)P(1), X(6)U(1), X(3053)U(2)

The perspector of the circumconic {{A,B,C,PU(39)}} is X(232).

The points PU(39) are isogonal conjugates of the points PU(37). P(39) is the {P(1),P(2)}-harmonic conjugate of X(3); also, U(39) is the {U(1),U(2)}-harmonic conjugate of X(3). (Randy Hutson, September 10, 2012)

#### P(40) = ISOGONAL CONJUGATE OF 1st BELTRAMI POINT

Trilinears    1/[a(b2 - a2)] : 1/[b(c2 - b2)] : 1/[c(a2 - c2)]
Barycentrics    1/(b2 - a2) : 1/(c2 - b2) : 1/(a2 - c2)

P(40) and U(40) lie on the central line X(115)X(125) and on the Steiner circumellipse.
P(40) lies on these non-central lines: P(133)U(136), U(91)U(133).
P(40) lies on these non-central lines: U(45)U(168), X(98)U(109), X(99)U(105), X(190)U(79), X(2787)U(27), X(2966)U(2), X(3568)P(2), U(1)U(38), U(131)U(135) (Randy Hutson, July 31, 2018)

P(40) = 2nd isobaryc of X(99)
P(40) = trilinear pole of line X(2)U(11)

#### P(41) = TRILINEAR PRODUCT U(7)*P(24)

Trilinears    (a - b)c/a : (b - c)/b : (c - a)/c
Barycentrics    (a - b)c : (b - c) : (c - a)

P(41) and U(41) lie on the line at infinity, X(30)X(511).

P(41) lies on these non-central lines: X(190)U(33), X(239)P(33), X(894)P(31), X(1278)U(53), X(3644)P(53), X(4360)U(31). (Randy Hutson, July 31, 2018)

It is easy to check that if P,U and P',U' are two bicentric pairs, then the trilinear products P*P', U*U' are either a pair of triangle centers or else a pair of bicentric points (and likewise for P*U', P'*U).

P(41) = crossdifference of every pair of points on line X(6)U(9)
U(41) = crossdifference of every pair of points on line X(6)P(9)

#### P(42) = ISOGONAL CONJUGATE OF P(41)

Trilinears    a2b/(a - b) : b2c/(b - c) : c2a/(c - a)
Barycentrics    a3b/(a - b) : b3c/(b - c) : c3a/(c - a)

P(42) and U(42) lie on the circumcircle.

P(42) and U(42) lie on the central line X(890)X(1977).

P(42) = trilinear pole of line X(6)U(9)
U(42) = trilinear pole of line X(6)P(9)

Index tri prd bary prd bic sum bic diff crsum crdiff tri pole ideal pt midpt cevapt crspt vertcon
37 304 69 3 525 1968 232 287 525 3 . . .
38 1969 264 5 525 1970 1971 1972 525 5 . . .
39 1973 1974 32 512 1975 325 1976 512 32 . . .
40 1577 523 690 115 . 110 523 690 115 . . .
41 1978 668 536 513 1979 6 2 . . . . .
42 1919 1980 890 1977 . 668 667 . . . . .

#### P(43) = 1st ACACIA POINT

Trilinears    bc[a2(a2 + 2b2 + 3c2) + b2(c2 - b2)] : :
Barycentrics    a2(a2 + 2b2 + 3c2) + b2(c2 - b2) : :

P(43) is the centroid of the pedal triangle of the 1st Brocard point, P(1).
The midpoint of PU(43) is X(9300).
P(43) and U(43) lie on the central line X(523)X(9300).
P(43) lies on these non-central lines: X(7736)P(45), X(7774)U(11), X(11174)P(11). (Randy Hutson, July 31, 2018)

P(43) lies on these non-central lines: X(2)U(179), P(1)P(186), U(176)U(186). (Randy Hutson, August 29, 2018)

#### P(44) = 1st LAEMMEL POINT

Trilinears    sin B + (cos A sin C - sin A) cos B : :
Barycentrics    (sin A)[sin B + (cos A sin C - sin A) cos B] : :

R. Laemmel posed the following problem, for which P(44) is the unique solution:

If from a point P is the plane of a triangle ABC, perpendiculars PA', PB', PC' to the sides BC, CA, AB are drawn, we get the segments BA', CB', AC'. Is there a point P for which these segments are of equal length? How long are they in this case?

For a discussion, see Darij Grinberg's Hyacinthos messages 7088 and 7089, May 4, 2003.

P(44) and U(44) lie on the central line X(3)X(667).

#### P(45) = ISOTOMIC CONJUGATE OF P(37)

Trilinears    c cos B : a cos C : b cos A
Trilinears    sin A - sin(B - C) : :
Barycentrics    cot B : cot C : cot A
Barycentrics    a^2 - b^2 + c^2 : :

P(45) and U(45) lie on the central line X(6)X(523).
P(45) lies on these non-central lines: X(2)P(11), X(69)U(11), X(193)U(132), X(371)U(161), X(372)P(161), X(514)U(46), X(522)P(46), X(877)U(37), X(4777)P(54), X(4802)U(54), X(7736)P(43), X(9131)U(62), X(9979)P(62), X(18581)P(5), X(18582)U(5), X(18912)P(61) P(1)P(37) U(6)P(47) U(31)U(47). (Randy Hutson, July 31, 2018)

P(45) lies on these non-central lines: X(15)P(175), X(16)U(175), X(5318)P(174), X(5321)U(174), U(40)P(168), U(137)U(181). (Randy Hutson, August 29, 2018)

P(45) and U(45) are a pair of polar conjugates.

P(45) = P(1)-Ceva conjugate of X(2)
P(45) = perspector of the cevian triangle of P(1) and the anticomplementary triangle
P(45) = 1st isobaryc of X(69)
P(45) = complement of P(132)
P(45) = anticomplement of P(11)
The perspector of the circumconic {{A,B,C,PU(45)}} is X(297).

See ADGEOM #63 and related postings, 5/15/2013.

#### P(46) = 1st BICENTRIC OF X(9)

Trilinears    a - b + c : b - c + a : c - a + b
Barycentrics    a(a - b + c) : b(b - c + a) : c(c - a + b)

P(46) and U(46) lie on the central line X(6)X(513).
P(46) lies on these non-central lines: X(1)P(6), X(9)U(6), X(37)U(55), X(44)U(56), X(514)U(45), X(522)P(45), X(883)P(47), X(1025)P(48), X(1100)P(55), X(1449)U(31), X(1743)P(33), X(3247)P(52), X(3729)P(41), X(3731)U(52), X(3875)U(41), X(4394)U(49), X(7290)U(96), X(9811)P(62), X(9810)U(62), X(16670)U(50), P(1)U(44), P(8)P(23) P(15)P(127) P(16)P(22) P(18)U(102) P(39)P(44) (Randy Hutson, July 31, 2018)

The perspector of the circumconic {{A,B,C,PU(46)}} is X(241).

#### P(47) = TRILINEAR PRODUCT X(2)*P(46)

Trilinears    bc(a - b + c) : ca(b - c + a) : ab(c - a + b)
Barycentrics    a - b + c : b - c + a : c - a + b

P(47) and U(47) lie on the central line X(1)X(514).
P(47) lies on these non-central lines: P(1)U(27), X(2)P(10), X(8)U(10), X(145)U(24), X(523)U(55), X(664)U(93), X(883)P(46), X(4977)P(55), X(9312)P(115) U(6)P(45) P(20)P(131) P(31)U(45). (Randy Hutson, July 31, 2018)

P(47) = isogonal conjugate of P(93); U(47) = isogonal conjugate of U(93)
P(47) = 1st isobaryc of X(8)

#### P(48) = TRILINEAR PRODUCT X(6)*P(46)

Trilinears    (a - b + c)a : (b - c + a)b : (c - a + b)c
Barycentrics    (a - b + c)a^2 : (b - c + a)b^2 : (c - a + b)c^2

P(48) and U(48) lie on the central line X(31)X(649).
The perspector of the circumconic {{A,B,C,PU(48)}} is X(1458).

#### P(49) = 1st BICENTRIC OF X(514)

Trilinears    (c - a)/b : (a - b)/c : (b - c)/a
Barycentrics    a(c - a)/b : b(a - b)/c : c(b - c)/a

P(49) and U(49) lie on the central line X(663)X(672).
P(49) lies on these non-central lines: X(238)P(26), P(1)P(9)

#### P(50) = 1st BICENTRIC OF X(44)

Trilinears    a - 2b + c : b - 2c + a : c - 2a + b
Barycentrics    a(a - 2b + c) : b(b - 2c + a) : c(c - 2a + b)

P(50) and U(50) lie on the central line X(44)X(513).
P(50) lies on these non-central lines: X(1)P(6), X(6)P(53), P(52)U(55), X(9)U(53), X(45)U(6), X(3246)U(96), X(4618)U(28), X(16666)U(31), X(16670)U(46), X(16672)P(52), X(16676)U(52) (Randy Hutson, July 31, 2018)

The perspector of the circumconic {{A,B,C,PU(50)}} is X(88).

Index tri prd bary prd bic sum bic diff crsum crdiff tri pole ideal pt midpt cevapt crspt vertcon
43 . . . 523 . . . 523 . . . .
44 . . 3 . . . . . 3 . . .
45 92 4 6 523 . 511 98 523 6 . . .
46 57 56 6 513 1376 518 105 513 6 . . .
47 85 7 1 514 . 672 673 514 1 . . .
48 604 1397 31 649 3729 3912 1438 . . . . .
49 101 692 672 663 . . . . . . . .
50 88 106 44 513 . 1 1 513 44 . . .

#### P(51) = 1st ACANTHUS POINT

Trilinears    2 - (b - c)/a : 2 - (c - a)/b : 2 - (a - b)/2
Barycentrics    2a - b + c : :

P(51) and U(51) lie on the central line X(1)X(514).

#### P(52) = 1st AMARANTH POINT

Trilinears    2b + c : 2c + a : 2a + b
Barycentrics    a(2b + c) : b(2c + a) : c(2a + b)

P(52) and U(52) lie on the central line X(37)X(513).
P(52) lies on these non-central lines: X(45)P(33), P(50)U(55)

#### P(53) = 1st ANGLICA POINT

Trilinears    2b - c : 2c - a : 2a - b
Barycentrics    a(2b - c) : b(2c - a) : c(2a - b)

P(53) and U(53) lie on the central line X(37)X(513).
P(53) lies on these non-central lines: X(6)P(50), X(9)U(50), X(1278)U(41), X(1449)U(31), X(1743)U(33), X(3644)P(41), X(16668)P(54), X(16671)U(54), X(16884)P(31), X(16885)P(33) (Randy Hutson, July 31, 2018)

#### P(54) = 1st ANTHERICUM POINT

Trilinears    2a - b + c : 2b - c + a : 2c - a + b
Barycentrics    a(2a - b + c) : b(2b - c + a) : c(2c - a + b)

P(54) and U(54) lie on the central line X(6)X(513).
P(54) lies on these non-central lines: X(1)P(56), X(9)U(55), X(37)P(31), X(44)U(6), X(1100)P(6), X(1449)P(55), X(1743)U(56), X(3723)P(52), X(4777)P(45), X(4802)U(45), X(16666)U(31), X(16668)P(53), X(16669)P(33), X(16671)U(53), X(16814)U(52) (Randy Hutson, July 31, 2018)

#### P(55) = 1st ARUM POINT

Trilinears    a + 2b : b + 2c : c + 2a
Barycentrics    a(a + 2b) : b(b + 2c) : c(c + 2a)

P(55) and U(55) lie on the central line X(1)X(513).
P(55) lies on the these non-central lines: X(6)P(6), X(9)U(54), X(37)U(46), X(45)P(33), X(1449)P(54), U(50)U(52).

#### P(56) = 1st ASH POINT

Trilinears    a - 2b : b - 2c : c - 2a
Barycentrics    a(a - 2b) : b(b - 2c) : c(c - 2a)

P(56) and U(56) lie on the central line X(9)X(513).
P(56) lies on these non-central lines: X(6)P(6), X(37)P(46)

#### P(57) = 1st BICENTRIC OF X(650)

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

P(57) and U(57) lie on the central line X(44)X(513).

#### P(58) = 1st BICENTRIC OF X(649)

Trilinears    1/c - 1/a : 1/a - 1/b : 1/b - 1/c
Barycentrics    1 - a/c : 1 - b/a : 1 - c/b

P(58) and U(58) lie on the central line X(44)X(513).
P(58) lies on non-central line X(239)P(24).

#### P(59) = 1st BICENTRIC OF X(56)

Trilinears    1 - cos B : 1 - cos C : 1 - cos A
Barycentrics    (sin A)(1 - cos B) : (sin B)(1 - cos C) : (sin C)(1 - cos A)

P(59) and U(59) lie on the central line X(650)X(3057).
P(59) = isogonal conjugate of P(92)

#### P(60) = 1st BICENTRIC OF X(40)

Trilinears    cos C + cos A - cos B - 1 : :
Barycentrics    (sin A)(cos C + cos A - cos B - 1) : :

P(60) and U(60) lie on the central line X(56)X(650).
P(60) lies on non-central line U(142)U(144).

#### P(61) = 1st VACARETU POINT

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

P(61) and U(61) occur in Daniel Vacaretu's work on left and right isoscelizers. The two points are closely related to the Rigby orthopole, X(1594). (D. Vacaretu, 10/17/03)

P(61) and U(61) lie on the central line X(523)X(1594).
P(61) lies on these non-central lines: X(7999)U(11), X(18912)P(45)

Index tri prd bary prd bic sum bic diff crsum crdiff tri pole ideal pt midpt cevapt crspt vertcon
51 . . 1 514 . 672 673 514 1 . . .
52 . . 37 513 . 238 291 513 37 . . .
53 . . 37 513 . 238 291 513 37 . . .
54 . . 6 513 . 518 105 513 6 . . .
55 . . 1 513 . 44 88 513 1 . . .
56 . . 9 513 . 1279 1280 513 9 . . .
57 651 109 1155 650 . 1 1 513 . . . .
58 190 100 899 649 . 1 1 513 . . . .
59 8 9 3057 650 . . . . . . . .
60 84 1436 56 650 . . . . . . . .
61 . . 1594 523 . . . 523 1594 . . .

#### P(62) = INSIMILICENTER OF 2nd LEMOINE AND PARRY CIRCLES

Trilinears    a(a2 - b2)(a2 + c2 - 2b2) : :
Barycentrics    a2(a2 - b2)(a2 + c2 - 2b2) : :

P(62) and U(62) are the internal and external centers of similitude of the 2nd Lemoine circle and the Parry circle. (Peter J. C. Moses, 10/22/03)

PU(62) lie on the central line X(6)X(351). (Randy Hutson, November 22, 2014)
P(62) lies on these non-central lines: X(110)P(105), X(111)U(107), X(371)U(63), X(372)P(63), X(485)U(65), X(486)P(65), X(1124)U(64), X(1335)P(64), X(1377)U(66), X(1378)P(66), X(1504)U(67), X(1505)P(67), X(7999)U(11), X(9131)U(45), X(9218)P(2), X(9810)U(46), X(9811)P(46), X(9979)P(45), X(18912)P(45) (Randy Hutson, July 31, 2018)

#### P(63) = INSIMILICENTER OF CIRCUMCIRCLE AND PARRY CIRCLE

Trilinears    a[(a2 - b2 - c2)S(a,b,c) - 4(b2 - c2)( b2 + c2 - 2a2)σ] : : ,
where S(a,b,c) = a4 + b4 + c4 - b2c2 - c2 a2 - a2b2, and σ = area(ABC)

P(63) and U(63) are the internal and external centers of similitude of the circumcircle and the Parry circle. (Peter J. C. Moses, 10/22/03)

PU(63) lie on the line X(3)X(351). (Randy Hutson, November 22, 2014)

#### P(64) = INSIMILICENTER OF INCIRCLE AND PARRY CIRCLE

Trilinears    a[bcS(a,b,c) - 2(b2 - c2)(2a2 - b2 - c2)σ] : : ,
where S(a,b,c) = a4 + b4 + c4 - b2c2 - c2 a2 - a2b2, and σ = area(ABC)

P(64) and U(64) are the internal and external centers of similitude of the incircle and the Parry circle. (Peter J. C. Moses, 10/22/03)

PU(64) lie on the line X(1)X(351). (Randy Hutson, November 22, 2014)

#### P(65) = INSIMILICENTER OF NINE-POINT AND PARRY CIRCLES

Trilinears    bc{[a2(b2 + c2) - (b2 - c2)2]S(a,b,c) - 4a2(b2 - c2)(2a2 - b2 - c2)σ} : : ,
where S(a,b,c) = a4 + b4 + c4 - b2c2 - c2 a2 - a2b2, and σ = area(ABC)

P(65) and U(65) are the internal and external centers of similitude of the nine-point circle and the Parry circle. (Peter J. C. Moses, 10/22/03)

PU(65) lie on the line X(5)X(351). (Randy Hutson, November 22, 2014)

#### P(66) = INSIMILICENTER OF SPIEKER AND PARRY CIRCLES

Trilinears    bc{a(b + c)[bcS(a,b,c) - 2a(b - c)(2a2 - b2 - c2)σ]},
where S(a,b,c) = a4 + b4 + c4 - b2c2 - c2 a2 - a2b2, and σ = area(ABC)

P(66) and U(66) are the internal and external centers of similitude of the Spieker circle and the Parry circle. (Peter J. C. Moses, 10/22/03)

PU(66) lie on the line X(10)X(351). (Randy Hutson, November 22, 2014)

#### P(67) = INSIMILICENTER OF MOSES AND PARRY CIRCLES

Trilinears    bc{a2(b2 + c2)S(a,b,c) - 4a2(b2 - c2)(2a2 - b2 - c2)σ},
where S(a,b,c) = a4 + b4 + c4 - b2c2 - c2 a2 - a2b2, and σ = area(ABC)

P(67) and U(67) are the internal and external centers of similitude of the Moses circle and the Parry circle. (Peter J. C. Moses, 10/22/03)

PU(67) lie on the line X(39)X(351). (Randy Hutson, November 22, 2014)

Index tri prd bary prd bic sum bic diff crsum crdiff tri pole ideal pt midpt cevapt crspt vertcon
62 . . . 351 . 543 . . . . . .
63 . . 3 351 . 10418 . . . . . .
64 . . . . . . . . . . . .
65 . . 5 351 . . . . . . . .
66 . . 10 351 . . . . . . . .
67 . . . . . . . . . . . .

#### P(68) = 1st BICENTRIC OF X(5)

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

P(68) and U(68) lie on the central line X(654)X(2594).
P(68) lies on these non-central lines: P(8)P(15) (the 1st bicentric of the Euler line), P(69)P(71)

The perspector of the circumconic {{A,B,C,PU(68)}} is X(2602).
P(68) = isogonal conjugate of P(69)

#### P(69) = 1st BICENTRIC OF X(54)

Trilinears    sec(C - A) : sec(A - B) : sec(B - C)
Barycentrics    sin A sec(C - A) : sin B sec(A - B) : sin C sec(B - C)

P(69) and U(69) lie on the central line X(2599)X(2600).
P(69) lies on non-central line P(68)P(71).

The perspector of the circumconic {{A,B,C,PU(69)}} is X(2596).
P(69) = isogonal conjugate of P(68)

#### P(70) = 1st BICENTRIC OF X(523)

Trilinears    sin(C - A) : sin(A - B) : sin(B - C)
Trilinears    c a (c^2 - a^2) : :
Barycentrics    sin A sin(C - A) : sin B sin(A - B) : sin C sin(B - C)

P(70) and U(70) lie on the central line X(2245)X(2605).
P(70) lies on non-central line X(238)P(26) (the 1st bicentric of the line at infinity).

X(2613) = perspector of conic {{A,B,C,PU(70)}}
P(70) = isogonal conjugate of P(71)

#### P(71) = 1st BICENTRIC OF X(110)

Trilinears    csc(C - A) : csc(A - B) : csc(B - C)
Barycentrics    sin A csc(C - A) : sin B csc(A - B) : sin C csc(B - C)

P(71) and U(71) lie on the central line X(2610)X(2611).
P(71) lies on these non-central lines: P(68)P(69), P(105)P(134)
The perspector of the circumconic {{A,B,C,PU(71)}} is X(2607).

P(71) = trilinear pole of (1st-bicentric-of-Brocard-axis = P(6)P(15)), which passes through X(2607)
U(71) = trilinear pole of (2nd-bicentric-of-Brocard-axis = U(6)U(15)), which passes through X(2607)
P(71) = isogonal conjugate of P(70)

#### P(72) = 1st BICENTRIC OF X(2616)

Trilinears    tan(C - A) : tan(A - B) : tan(B - C)
Barycentrics    sin A tan(C - A) : sin B tan(A - B) : sin C tan(B - C)

P(72) and U(72) lie on the central line X(2290)X(2618).
The perspector of the circumconic {{A,B,C,PU(72)}} is X(2626).
P(72) = isogonal conjugate of P(73)

#### P(73) = 1st BICENTRIC OF X(2617)

Trilinears    cot(C - A) : cot(A - B) : cot(B - C)
Barycentrics    sin A cot(C - A) : sin B cot(A - B) : sin C cot(B - C)

P(73) and U(73) lie on the central line X(1109)X(2624).>br> P(73) = isogonal conjugate of P(72)

#### P(74) = 1st BICENTRIC OF X(656)

Trilinears    tan C - tan A : tan A - tan B : tan B - tan C
Barycentrics    (sin A)(tan C - tan A) : (sin B)(tan A - tan B) : (sin C)(tan B - tan C)

P(74) and U(74) lie on the central line X(44)X(513).
P(74) = isogonal conjugate of P(75)

#### P(75) = 1st BICENTRIC OF X(162)

Trilinears    1/(tan C - tan A) : 1/(tan A - tan B) : 1/(tan B - tan C)
Barycentrics    (sin A)/(tan C - tan A) : (sin B)/(tan A - tan B) : (sin C)/(tan B - tan C)

P(75) and U(75) lie on the central line X(2631)X(2632).
P(75) = isogonal conjugate of P(74)

Index tri prd bary prd bic sum bic diff crsum crdiff tri pole ideal pt midpt cevapt crspt vertcon
68 54 2148 2594 654 2595 2596 2597 . 2598 . . .
69 5 1953 2599 2600 2601 2602 2603 . 2604 . . .
70 110 163 2245 2605 2606 2607 2608 . 2609 . . .
71 523 661 2610 2611 2612 2613 2614 . 2615 . . .
72 2617 1625 2290 2618 2619 2620 2621 . 2622 . 2625 .
73 2616 2623 2624 1109 2625 2626 2627 . 2628 . . .
74 162 112 2173 656 2629 1 1 513 2630 . . .
75 656 647 2631 2632 2633 162 656 . 2634 . . .

#### P(76) = 1st BICENTRIC OF X(652)

Trilinears    sec C - sec A : :
Barycentrics    (sin A)(sec C - sec A) : :

P(76) and U(76) lie on the central line X(44)X(513).
P(76) = isogonal conjugate of P(77)

#### P(77) = 1st BICENTRIC OF X(653)

Trilinears    1/(sec C - sec A) : :
Barycentrics    (sin A)/(sec C - sec A) : :

P(77) and U(77) lie on the central line X(2637)X(2638).
P(77) = isogonal conjugate of P(76)

#### P(78) = 1st BICENTRIC OF X(661)

Trilinears    cot C - cot A : :
Trilinears    c^2 - a^2 : :
Barycentrics    (sin A)(cot C - cot A) : :

P(78) and U(78) lie on the central line X(44)X(513).
P(78) = isogonal conjugate of P(79)

#### P(79) = 1st BICENTRIC OF X(662)

Trilinears    1/(cot C - cot A) : :
Barycentrics    (sin A)/(cot C - cot A) : :

P(79) and U(79) lie on the central line X(2642)X(2643).
P(79) = isogonal conjugate of P(78)

#### P(80) = 1st BICENTRIC OF X(65)

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

P(80) and U(80) lie on the central line X(650)X(2646).
P(80) = isogonal conjugate of P(81)

#### P(81) = 1st BICENTRIC OF X(21)

Trilinears    1/(cos C + cos A) : :
Barycentrics    (sin A)/(cos C + cos A) : :

P(81) and U(81) lie on the central line X(661)X(2650).
P(81) lies on non-central line P(8)P(15) (the 1st bicentric of the Euler line).
P(81) = isogonal conjugate of P(80)

#### P(82) = 1st BICENTRIC OF X(73)

Trilinears    sec C + sec A : :
Barycentrics    (sin A)(sec C + sec A) : :

P(82) and U(82) lie on the central line X(652)X(2654).
P(82) = isogonal conjugate of P(83)

#### P(83) = 1st BICENTRIC OF X(29)

Trilinears    1/(sec C + sec A) : :
Barycentrics    (sin A)/(sec C + sec A) : :

P(83) and U(83) lie on the central line X(822)X(2658).
P(83) = isogonal conjugate of P(82)

#### P(84) = 1st BICENTRIC OF X(42)

Trilinears    1/c + 1/a : :
Barycentrics    1 + a/c : :

P(84) and U(84) lie on the central line X(649)X(2666).
P(84) lies on these non-central lines: X(1)P(8), P(32)P(34).
P(84) = isogonal conjugate of P(85)

#### P(85) = 1st BICENTRIC OF X(86)

Trilinears    ac/(a + c) : :
Barycentrics    a2c/(a + c) : :

P(85) and U(85) lie on the central line X(798)X(2667).
P(85) = isogonal conjugate of P(84)

Index tri prd bary prd bic sum bic diff crsum crdiff tri pole ideal pt midpt cevapt crspt vertcon
76 653 108 2635 652 2636 1 1 513 . . . .
77 652 . 2637 2638 2639 653 652 . . . . .
78 662 110 896 661 2640 1 1 513 2641 . . .
79 661 512 2642 2643 2644 662 661 . 2645 . . .
80 21 284 2646 650 2647 1758 2648 . 2649 . . .
81 65 1400 2650 661 409 2651 2652 . 2653 . . .
82 29 1172 2654 652 2662 2655 2656 . 2657 . . .
83 73 1409 2658 822 410 2659 2660 . 2661 . . .
84 86 81 3720 649 2663 2664 2665 . 2666 . . .
85 42 213 2667 798 2668 2669 2107 . 2670 . . .

#### P(86) = 1st BICENTRIC OF X(30)

Trilinears    ca[2b4 - (c2 -a2)2 - b2(c2 + a2)] : :

The points P(86) and U(86) lie on the cenrtral line X(1464)X(9404).
P(86) lies on these non-central lines: X(238)P(26) (the 1st bicentric of the line at infinity), and P(8)P(15) (the 1st bicentric of the Euler line).

#### P(87) = TRILINEAR QUOTIENT P(9)/P(86)

Trilinears    a/[2b4 - (c2 -a2)2 - b2(c2 + a2)] : :

P(87) and U(87) lie on the circumcircle.

#### P(88) = 1st BICENTRIC OF X(511)

Trilinears    b(b2c2 + b2a2 - c4 - a4) : :

The points PU(88), U(88) and lie on the line X(1284)X(3287).
P(88) lies on these non-central lines: X(238)P(26) (the 1st bicentric of the line at infinity), X(2607)P(6) (the 1st bicentric of the Brocard axis), X(5270)P(16).

#### P(89) = TRILINEAR QUOTIENT P(9)/P(88)

Trilinears    a/[b2(b2c2 + b2a2 - c4 - a4)] : :

P(89) and U(89) lie on the circumcircle and the line X(2491)X(9419).
The perspector of the circumconic {{A,B,C,PU(89)}} is X(6).

The lines PU(89) and PU(109) are parallel.
P(89) lies on these non-central lines: X(74)P(187), X(2698)P(2), X(21444)U(91), X(22456)U(177), U(1)P(109), P(39)P(91), P(105)U(157), P(133)P(169)
P(89) = isogonal conjugate of P(177)

#### P(90) = 1st BICENTRIC OF X(512)

Trilinears    b(c2 - a2) : :

P(90) and U(90) lie on the central line X(2238)X(4367).
P(90) lies on these non-central lines: X(239)P(24),, X(238)P(26) (the 1st bicentric of the line at infinity), U(9)U(10) .

#### P(91) = TRILINEAR QUOTIENT P(9)/P(90)

Trilinears    a/(b2(c2 - a2)) : :

P(91) = barycentric product of circumcircle intercepts of line X(2)P(1)
P(91) and U(91) lie on the circumcircle and the line X(887)X(1084).
P(91) lies on these non-central lines: X(805)P(2), P(1)U(133), U(40)P(133).

P(91) = trilinear pole of line X(6)U(148)

#### P(92) = 1st BICENTRIC OF X(8)

Trilinears    (a - b + c)/b : :
Barycentrics    (a - b + c)a/b : :

P(92) and U(92) lie on the central line X(649)X(1201).
P(92) lies on these non-central lines: X(1)P(8), X(1083)P(1).
P(92) = isogonal conjugate of P(59)

#### P(93) = TRILINEAR QUOTIENT P(9)/P(92)

Trilinears    a/(a - b + c) : :
Barycentrics    a2/(a - b + c) : :

P(93) and U(93) lie on the central line X(41)X(663). PU(93) are the isogonal conjugates of PU(47).

#### P(94) = 1st BICENTRIC OF X(55)

Trilinears    (a - b + c)b : :
Barycentrics    a(a - b + c)b : :

P(94) and U(94) lie on the central line X(354)X(650).
P(94) lies on non-central line U(142)U(143)

#### P(95) = TRILINEAR QUOTIENT P(9)/P(94)

Trilinears    a/[b2(a - b + c) : :
Barycentrics    a2/[b2(a - b + c) : :

P(95), U(95), and X(8638) are collinear.

#### P(96) = 1st BICENTRIC OF X(518)

Trilinears    bc + ba - c2 - a2 : :
Barycentrics    a(bc + ba - c2 - a2) : :

P(96) and U(96) lie on the central line X(513)X(663).
P(96) lies on these non-central lines: X(1)P(6), X(238)P(26) (the 1st bicentric of the line at infinity)

The perspector of the circumconic {{A,B,C,PU(96)}} is X(294).

#### P(97) = TRILINEAR QUOTIENT P(9)/P(96)

Trilinears    a/(b(bc + ba - c2 - a2)) : :
Barycentrics    a2/(b(bc + ba - c2 - a2)) : :

P(97) and U(97) lie on the circumcircle.

#### P(98) = 1st BICENTRIC OF X(519)

Trilinears    ca(2b - c - a) : :
Barycentrics    ca2(2b - c - a)

P(98) and U(98) lie on the central line X(649)X(1149)
P(98) lies on these non-central lines: X(1)P(8), X(238)P(26) (the 1st bicentric of the line at infinity)

#### P(99) = TRILINEAR QUOTIENT P(9)/P(98)

Trilinears    a2/(2b - c - a) : :
Barycentrics    a/(2b - c - a)

P(99) and U(99) lie on the circumcircle and the central line X(1017)X(1960).

Index tri prd bary prd bic sum bic diff crsum crdiff tri pole ideal pt midpt cevapt crspt vertcon
86 74 2159 1464 bic diff crosssum crossdiff tril pole ideal mid . . .
87 tri prod bary prod bic sum bic diff crosssum crossdiff tril pole ideal mid . . .
88 98 1910 1284 3287 crosssum crossdiff tril pole ideal mid cevapt crspt vertcon
89 tri prod bary prod bic sum bic diff crosssum 290 237 ideal mid cevapt crspt vertcon
90 99 662 2238 4367 7170 3571 tril pole ideal mid cevapt crspt vertcon
91 tri prod bary prod bic sum bic diff crosssum crossdiff tril pole ideal mid cevapt crspt vertcon
92 tri prod bary prod bic sum bic diff crosssum crossdiff tril pole ideal mid cevapt crspt vertcon
93 tri prod bary prod bic sum bic diff crosssum crossdiff tril pole ideal mid cevapt crspt vertcon
94 tri prod bary prod bic sum bic diff crosssum crossdiff tril pole ideal mid cevapt crspt vertcon
95 tri prod bary prod bic sum bic diff crosssum crossdiff tril pole ideal mid cevapt crspt vertcon
96 tri prod bary prod 1279 513 crosssum 9 57 513 1279 cevapt crspt vertcon
97 tri prod bary prod bic sum bic diff crosssum crossdiff tril pole ideal mid cevapt crspt vertcon
98 106 bary prod 1149 649 crosssum crossdiff tril pole ideal mid cevapt crspt vertcon
99 tri prod bary prod bic sum bic diff crosssum crossdiff tril pole ideal mid cevapt crspt vertcon

#### P(100) = 1st BICENTRIC OF X(521)

Trilinears    (sec C - sec A)/b : :
Barycentrics    a(sec C - sec A)/b : :

P(100) and U(100) lie on the central line X(2182)X(6129).
P(100) lies on non-central line X(238)P(26) (the 1st bicentric of the line at infinity).

#### P(101) = TRILINEAR QUOTIENT P(9)/P(100)

Trilinears    a/(sec C - sec A) : :
Barycentrics    a2/(sec C - sec A) : :

P(101) and U(101) lie on the circumcircle.

P(101) lies on these non-central lines: X(100)P(77), X(2714)P(2)
P(101) = trilinear pole of line X(6)P(19)

#### P(102) = 1st BICENTRIC OF X(522)

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

P(102) and U(102) lie on the central line X(1459)X(2183).
P(102) lies on non-central line X(238)P(26) (the 1st bicentric of the line at infinity).

#### P(103) = TRILINEAR QUOTIENT P(9)/P(102)

Trilinears    a/(cos C - cos A) : :
Barycentrics    a2/(cos C - cos A) " "

P(103) and U(103) lie on the circumcircle and the central line X(3271)X(6139).

P(103) = trilinear pole of line X(6)P(18)

#### P(104) = 1st BICENTRIC OF X(7)

Trilinears    ca/(c + a - b) : :
Barycentrics    ca2/(c + a - b) : :

P(104) and U(104) lie on the central line X(657)X(2293)

#### P(105) = TRILINEAR QUOTIENT P(9)/P(70)

Trilinears    a/(c2 - a2) : :
Barycentrics    a2/(c2 - a2) : :

P(105) and U(105) lie on the circumcircle.
P(105) and U(105) lie on the central line X(351)X(865).
P(105) lies on these non-central lines: X(2)U(133), X(6)P(107), X(691)P(2), X(2502)U(107), P(32)P(34), P(71)P(134)

P(105) = trilinear pole of line X(6)U(1) (the tangent to the 2nd Brocard circle at U(1))
P(105) = trilinear quotient X(6)/P(78)
P(105) = isogonal conjugate of P(179)

#### P(106) = 1st BICENTRIC OF X(524)

Trilinears    (2b2 - c2 - a2)/b : :
Barycentrics    a(2b2 - c2 - a2)/b

P(106) lies on non-central line X(238)P(26) (the 1st bicentric of the line at infinity).

#### P(107) = TRILINEAR QUOTIENT P(9)/P(106)

Trilinears    a/(2b2 - c2 - a2) : :
Barycentrics    a2/(2b2 - c2 - a2) : :

P(107) and U(107) lie on the circumcircle and are collinear with X(351).
P(107) lies on these non-central lines: X(6)P(105), X(2502)U(105).

P(107) = trilinear pole of line X(6)U(2)
P(107) = isogonal conjugate of P(180)

#### P(108) = 1st BICENTRIC OF X(525)

Trilinears    (c cos C - a cos A)/b : :
Barycentrics    a(c cos C - a cos A)/b : :

P(108) lies on non-central line X(238)P(26) (the 1st bicentric of the line at infinity).

#### P(109) = TRILINEAR QUOTIENT P(9)/P(108)

Trilinears    a/(c cos C - a cos A) : :
Barycentrics    a2/(c cos C - a cos A) : :

P(109) and U(109) lie on the circumcircle.
The lines PU(89) and PU(109) are parallel.
P(109) lies on non-central line P(37)U(133).
P(109) and U(109) line on central line X(3269)X(9409).

P(109) = Λ(trilinear polar of P(45))
P(109) = trilinear pole of line X(6)U(157)

#### P(110) = 1st MONTESDEOCA CIRCUMCIRCLES RADICAL CENTER

Trilinears    a2(b - 2c) - a(b2 + bc - c2) - (b - c)c2 : :
Barycentrics    a3(b - 2c) - a(b2 + bc - c2) - (b - c)c2 : :

Let A'B'C' be the cevian triangle of X(1). Let AB be the reflection of A' in BB', and define BC and CA cyclically. Let AC be the reflection of A' in BC', and define BA and CB cyclically. Let OAB be the circumcircle of AA'AB, and define OBC and OCA cyclically. Let OAC be the circumcircle of AA'AC, and define OBA and OCB cyclically. Then P(110) is the radical center of OAB, OBC, OCA, and the 2nd Montesdeoca circumcircles radical center, U(110) is defined symmetrically; i.e., as the radical center of OAC, OBA, OCB.    (Angel Montesdeoca, August 26 2013)

P(110) and U(110) lie on the line X(942)X(1938); P(110)U(110) has ideal point X(1938).

#### P(111) = 1st MONTESDEOCA NINE-POINT CIRCLES RADICAL CENTER

Trilinears    (a + b - c)(a - b + c)(a3b + 2a3c + a2c2 - ab3 - 2ac3 - 2ab2c - 3abc2 - 3bc3 - 3b2c2 - b3c - c4) : :

Let A'B'C' be the cevian triangle of X(1). Let NAB be the nine-point circle of AIB, where I = X(1), and define NBC and NCA cyclically. Let NAC be the nine-point circle of AIC, and define NBA and NCB cyclically. Then P(111) is the radical center of NAB, NBC, NCA, and the 2nd Montesdeoca nine-point circles radical center, U(111), is defined symmetrically; i.e., as the radical center of NAC, NBA, NCB.    (Angel Montesdeoca, August 26 2013)

The ideal point of P111) is X(513).

Index tri prd bary prd bic sum bic diff crsum crdiff tri pole ideal pt midpt cevapt crspt vertcon
100 108 bary prod 2182 6129 crosssum crossdiff tril pole ideal mid . . .
101 tri prod bary prod bic sum bic diff crosssum crossdiff tril pole ideal mid cevapt crspt vertcon
102 109 1415 2183 1459 crosssum crossdiff tril pole ideal mid cevapt crspt vertcon
103 tri prod bary prod bic sum bic diff crosssum crossdiff tril pole ideal mid . . .
104 55 41 2293 657 crosssum crossdiff tril pole ideal mid cevapt crspt vertcon
105 tri prod bary prod bic sum bic diff crosssum crossdiff tril pole ideal mid cevapt crspt vertcon
106 111 923 bic sum bic diff crosssum crossdiff tril pole ideal mid cevapt crspt vertcon
107 tri prod bary prod bic sum bic diff crosssum crossdiff tril pole ideal mid cevapt crspt vertcon
108 112 bary prod bic sum bic diff crosssum crossdiff tril pole ideal mid cevapt crspt vertcon
109 tri prod bary prod bic sum bic diff crosssum crossdiff tril pole ideal mid . . .
110 tri prod bary prod bic sum bic diff crosssum crossdiff tril pole ideal mid cevapt crspt vertcon
111 tri prod bary prod bic sum bic diff crosssum crossdiff tril pole ideal mid cevapt crspt vertcon

#### P(112) = 1st BICENTRIC OF X(57)

Trilinears    1/(a - b + c) : :
Barycentrics    a/(a - b + c)

PU(112) are the isogonal conjugates of PU(46). PU(112) lie on the conic {{A, B, C, X(2), X(100)}}. (Randy Hutson, September 29, 2014)

PU(112) lie on the line X(55)X(650)
The perspector of the circumconic {{A,B,C,PU(112)}} is X(518).

#### P(113) = 1st BICENTRIC QUOTIENT OF X(37)

Trilinears    (a + c)/(a + b) : :
Barycentrics    a(a + c)/(a + b) : :

Suppose that X = x : y : z (trilinears) is a triangle center. Define P(X) = y/z : z/x : x/y and U(X) = z/y : x/z : y/x, introduced here as the 1st and 2nd bicentric quotients of X, respectively. The pair is both a bicentric pair and an isogonal conjugate pair. As an example, P(X(2)) and U(X(2)) are the 1st and 2nd Brocard points.

The perspector of the circumconic {{A,B,C,PU(113)}} is X(6157).

P(113) and U(113) lie on central line X(4983)X(6155).

P(113) = 2nd bicentric quotient of X(81)

#### P(114) = 1st BICENTRIC QUOTIENT OF X(513)

Trilinears    (a - c)/(a - b) : :
Barycentrics    a(a - c)/(a - b) : :

See P(113) for the definition of bicentric quotient.

P(114) lies on these non-central lines: X(1)U(6), X(5091)P(1).

The perspector of the circumconic {{A,B,C,PU(114)}} is X(6163).

P(114) and U(114) lie on central line X(2087)X(6161).

P(114) = 2nd bicentric quotient of X(100)

#### P(115) = 1st BICENTRIC QUOTIENT OF X(9)

Trilinears    (a + c - b)/(a + b - c) : :
Barycentrics    a(a + c - b)/(a + b - c) : :

See P(113) for the definition of bicentric quotient.

The perspector of the circumconic {{A,B,C,PU(115)}} is X(6168).

P(115) and U(115) lie on central line X(663)X(2082).

P(115) = 2nd bicentric quotient of X(57)

#### P(116) = 1st REAL FOCUS OF STEINER CIRCUMELLIPSE

Barycentrics    2(b2 - c2)(a4 - b2c2 - a2Z) + (V - W)1/2 : : , where
Z = (a4 + b4 + c4 - b2c2 - c2a2 - a2b2)1/2
V = 2a2b2c2Z3
W = b6c6 + c6a6 + a6b6 - 3a4b4c4 - (b4c4 + c4a4 + a4b4)Z2

PU(116) are also known as the Bickart points. Contributed with coordinates by Peter Moses, January 6, 2015. See also P117-119. (The indicated barycentrics are for P(116); for U(116), the 2nd real focus, barycentrics are given by 2(b2 - c2)(a4 - b2c2 - a2Z) - (V - W)1/2

P(116) and U(116) lie on central line X(2)X(1341).

P(116) = anticomplement of U(118)
U(116) = anticomplement of P(118)

#### P(117) = 1st IMAGINARY FOCUS OF STEINER CIRCUMELLIPSE

Barycentrics    2(b2 - c2)(a4 - b2c2 + a2Z) + (- V - W)1/2 : : , where
Z = (a4 + b4 + c4 - b2c2 - c2a2 - a2b2)1/2
V = 2a2b2c2Z3
W = b6c6 + c6a6 + a6b6 - 3a4b4c4 - (b4c4 + c4a4 + a4b4)Z2

Contributed with coordinates by Peter Moses, January 6, 2015. See also P116. (The indicated barycentrics are for P(117); for U(117), the 2nd imaginary focus, barycentrics are given by 2(b2 - c2)(a4 - b2c2 + a2Z) - ( - V - W)1/2.

P(117): P(117) and U(117) lie on central line X(2)X(1340).

#### P(118) = 1st REAL FOCUS OF STEINER INELLIPSE

Barycentrics    (b2 - c2)(a4 - b2c2 - a2Z) + (V - W)1/2 : :, where
Z = (a4 + b4 + c4 - b2c2 - c2a2 - a2b2)1/2
V = 2a2b2c2Z3
W = b6c6 + c6a6 + a6b6 - 3a4b4c4 - (b4c4 + c4a4 + a4b4)Z2

Contributed with coordinates by Peter Moses, January 6, 2015. See also P116 and P119. (The indicated barycentrics are for P(118); for U(118), the 2nd real focus, barycentrics are given by (b2 - c2)(a4 - b2c2 - a2Z) - (V - W)1/2

P(118) and U(118) are the only real points which are the centroids of their antipedal triangles. Also, P(118) and U(118) are the only real isogonal conjugates whose midpoint is X(2). (Randy Hutson, April 9, 2016)

P(118) and U(118) lie on central line X(2)X(1341).

P(118) = complement of U(116)
U(118) = complement of P(116)

#### P(119) = 1st IMAGINARY FOCUS OF STEINER INELLIPSE

Barycentrics    (b2 - c2)(a4 - b2c2 + a2Z) + (- V - W)1/2 : : , where
Z = (a4 + b4 + c4 - b2c2 - c2a2 - a2b2)1/2
V = 2a2b2c2Z3
W = b6c6 + c6a6 + a6b6 - 3a4b4c4 - (b4c4 + c4a4 + a4b4)Z2

Contributed with coordinates by Peter Moses, January 6, 2015. See also P116-P118. (The indicated barycentrics are for P(119); for U(119), the 2nd imaginary focus, barycentrics are given by (b2 - c2)(a4 - b2c2 + a2Z) - ( - V - W)1/2

P(119) and U(119) lie on central line X(2)X(1340).

Index tri prd bary prd bic sum bic diff crsum crdiff tri pole ideal pt midpt cevapt crspt vertcon
112 tri prod bary prod bic sum bic diff crosssum 241 294 ideal mid . 1376 25
113 tri prod bary prod bic sum bic diff crosssum crossdiff tril pole ideal mid . . .
114 1 6 6161 2087 6162 6163 6164 6165 6166 . . .
115 1 6 2082 663 6167 6168 6169 6170 6171 . . .
116 tri prod bary prod 2 3413 crosssum 5639 tril pole 3413 2 . 3146 .
117 tri prod bary prod 2 3414 crosssum 5638 tril pole 3414 2 . 3146 .
118 1 6 2 3413 3557 5639 tril pole 3413 2 6177 3557 .
119 1 6 2 3414 3558 5638 tril pole 3414 2 6178 3558 .
120 tri prod bary prod bic sum bic diff crosssum crossdiff tril pole ideal mid . . .
121 tri prod bary prod bic sum bic diff crosssum crossdiff tril pole ideal mid . . .
122 tri prod bary prod bic sum bic diff crosssum crossdiff tril pole ideal mid . . .

#### P(120) =  1st INTERCEPT OF LINE X(7)X(812) AND ADAMS CIRCLE

Barycentrics    c(a - c)(a + b - c)(a - b + c)(a2 + b2 - ac - bc) : :

Contributed by Peter Moses, June 1, 2015.

P(120) and U(120) lie on central line X(7)X(812).
P(120) lies on non-central line X(1)U(122).

#### P(121) =  1st INTERCEPT OF LINE X(11)X(244) AND ADAMS CIRCLE

Barycentrics    (b - c)(a - b - c)(a3 + c3 - a2b + b2c - 2ac2 - 2bc2 + 2abc) : :

Contributed by Peter Moses, June 1, 2015.

P(121) and U(121) lie on central line X(11)X(244).

#### P(122) = 1st INTERCEPT OF LINE X(390)X(812) AND ADAMS CIRCLE

Barycentrics    (a-b-c) (a^2+b^2-a c-b c) (a^3 b-2 a^2 b^2+a b^3+2 a^3 c+a^2 b c+b^3 c-2 a^2 c^2-a b c^2-2 b^2 c^2+b c^3) : :

Contributed by Peter Moses, June 1, 2015.

P(122) and U(122) lie on central line X(390)X(812).
P(122) lies on non-central line X(1)U(120).

#### P(123) =  1st INTERCEPT OF LINE X(11)X(244) AND CONWAY CIRCLE

Barycentrics    (a-b-c) (a^2+b^2-a c-b c) (a^3 b-2 a^2 b^2+a b^3+2 a^3 c+a^2 b c+b^3 c-2 a^2 c^2-a b c^2-2 b^2 c^2+b c^3) : :

Contributed by Peter Moses, June 2, 2015.

P(123) and U(123) lie on central line X(11)X(244).

#### P(124) = 1st MONTESDEOCA TRILINEAR POLE

Trilinears    (b/a)1/3 : (c/b)1/3 : (a/c)1/3
Barycentrics    (ba2)1/3 : (cb2)1/3 : (ac2)1/3

Contributed by Angel Montesdeoca, October 9, 2015. At X(8183), a degenerate conic is defined by using k = -(a + b + c)a-1/3b-1/3c-1/3. The conic consists of two lines whose trilinear polars are PU(124).

The ideal point of PU(124) is X(513).

#### P(125) = 1st BICENTRIC OF X(46)

Trilinears    cos A + cos C - cos B : :
Trilinears    a^3-(c-b)*a^2-(b^2+c^2)*a-(b^2-c^2)*(b+c) : :
Barycentrics    (sin A)(cos A + cos C - cos B) : :

P(125) and U(125) lie on central line X(3)X(650).

#### P(126) = 1st BICENTRIC OF X(1745)

Trilinears    sec A + sec C - sec B : :
Trilinears    (b-c)*a^5-a^4*b*c-2*(b^3-c^3)*a^3+(b^2-c^2)*(b+c)*a*(b^2+c^2)+(b^2-c^2)^2*b*c : :
Barycentrics    (sin A)(sec A + sec C - sec B) : :

P(126) and U(126) lie on central line X(4)X(652).

#### P(127) = 1st BICENTRIC OF X(610)

Trilinears: P(127) a^4+2*(b^2-c^2)*a^2-(b^2-c^2)*(c^2+3*b^2) : :
Trilinears    tan A + tan C - tan B : :
Trilinears    a^4+2*(b^2-c^2)*a^2-(b^2-c^2)*(c^2+3*b^2) : :
Barycentrics    (sin A)(tan A + tan C - tan B) : :

P(127) and U(127) lie on central line X(19)X(656).

#### P(128) = 1st BICENTRIC OF X(1707)

Trilinears    cot A + cot C - cot B : :
Trilinears    3*b^2-c^2-a^2 : :
Barycentrics    (sin A)(cot A + cot C - cot B) : :

P(128) and U(128) lie on central line X(63)X(661).

#### P(129) = 1st ISOBARYC OF X(5905)

Trilinears    b c (a^3+a^2 b-a b^2-b^3-a^2 c-b^2 c-a c^2+b c^2+c^3) : :
Barycentrics    cos A + cos C - cos B : :

P(129) and U(129) lie on central line X(63)X(522).

#### P(130) = 1st ISOBARYC OF X(6360)

Trilinears    b c(b^5(c + a) + b^4c a - 2b^3(c^3 + a^3) + b(c - a)^2(c + a)(c^2 + a^2) - c a(c^2 - a^2)^2) : :
Barycentrics    sec A + sec C - sec B : :

P(130) and U(130) lie on central line X(92)X(521).

#### P(131) = 1st ISOBARYC OF X(20)

Trilinears    b c(-3b^4 + 2b^2(c^2 + a^2) + (c^2 - a^2)^2) : :
Barycentrics    tan A + tan C - tan B : :

P(131) and U(131) lie on central line X(4)X(525).
P(131) lies on non-central line X(2)P(37) (the 1st isobaryc of the Euler line).

#### P(132) = 1st ISOBARYC OF X(193)

Trilinears    bc(3b^2 - c^2 - a^2) : :
Barycentrics    cot A + cot C - cot B : :

P(132) and U(132) lie on central line X(69)X(523).
P(132) lies on these non-central lines: X(2)P(11), X(193)U(45), P(131)P(177), X(487)U(161), X(488)P(161), X(3620)U(11). (Randy Hutson, July 31, 2018)

P(132) = anticomplement of P(45)

Index tri prd bary prd bic sum bic diff crsum crdiff tri pole ideal pt midpt cevapt crspt vertcon
123 tri prod bary prod bic sum bic diff crosssum crossdiff tril pole ideal mid . . .
124 tri prod bary prod bic sum bic diff crosssum crossdiff tril pole ideal mid . . .
125 90 2164 3 650 8757 8758 8759 8760 mid . . .
126 3362 8761 4 652 8762 8763 8764 ideal mid . . .
127 2184 64 19 656 8765 8766 8767 8768 2 . . .
128 8769 8770 63 661 8771 8772 8773 8774 . . . .
129 . . 63 522 8775 8776 8777 522 63 . . .
130 . 7361 92 521 . . . 521 92 . . .
131 tri prod 253 4 525 8778 8779 6330 525 4 . . .
132 tri prod 2996 69 523 8780 1692 8781 523 69 . . .

### Bicentric Pairs PU(133)-PU(145) César Lozada, January 26, 2016

It is well known that if A'B'C' is perspective to both A"B"C" and to B"C"A", then A'B'C' is also perspective to C"A"B", and in this case, A'B'C' and A"B"C" are triple-perspective. The two perspectors are here named the BCA-perspector of A'B'C' and A''B''C'' and the CAB-perspector of A'B'C' and A''B''C''.

The appearance of (T1, T2, i, n) in the following list means that triangles T1 and T2 are triple-perspective, with X(i) = central perspector, P(n) = BCA-perspector, and U(n) = CAB-perspector, unless an asterisk, *, is included, and in that case U(n) = BCA-perspector, and P(n) = CAB-perspector.

(ABC, 1st anti-Brocard, 1916, 133)
(ABC, 1st Brocard, 76, 1*)
(ABC, 3rd Brocard, 32, 1)
(ABC, 2nd Sharygin, 291, 134)
(1st anti-Brocard, anticomplementary, 147, 135)
(1st anti-Brocard, medial, 8290, 136)
(anticomplementary, 1st Brocard, 2896, 137)
(anti-McCay, 1st Parry, 2, 138)
(1st Brocard, 3rd Brocard, 384, 1*)
(1st Brocard, medial, 3, 38)
(3rd Brocard, circumsymmedial, 6195, 139)
(3rd Brocard, symmedial, 194, 140)
(3rd Brocard, tangential, 3499, 141)
(excentral, 2nd Sharygin, 3499, 142)
(incentral, 2nd Sharygin, 8298, 143)
(2nd mixtilinear, 2nd Sharygin, 8285, 144)
(inner-Napoleon, outer-Napoleon, 3, 5 )
(2nd Parry, 3rd Parry, 647, 145)

Also the 2nd Brocard triangle is triple-perspective to the Lucas-Brocard and the Lucas-Brocard(-1) triangles defined at X(6421), but their BCA- and CAB- perspectors are rather complicated and are not written here.

#### P(133) = BCA-PERSPECTOR OF THESE TRIANGLES: ABC and 1st ANTI-BROCARD

Trilinears    bc/(a^2*b^2-c^4) : :
Barycentrics    1/(a^2*b^2-c^4) : :

P(133) and U(133) lie on the circumcircle of ABC and on the central line X(804)X(5976).
P(133) lies on these non-central lines: X(2)U(105), X(733)P(136), X(1281)U(134), U(1)U(91), U(37)U(109), P(40)U(136), U(40)P(91)

P(133) = P(1)-of-1st antiBrocard-triangle
P(133) = 2nd isobaryc of X(1916)
P(133) = trilinear pole of line X(6)P(11)

#### P(134) = BCA-PERSPECTOR OF THESE TRIANGLES: ABC and 2nd SHARYGIN

Trilinears    1/(a*b-c^2) : :
Barycentrics    a/(a*b-c^2) : :

P(134) and U(134) lie on the circumcircle of ABC and on the central line X(659)X(812).
P(134) lies on these non-central lines: X(1)P(25), X(1281)U(133), U(32)U(34), P(71)P(105)

P(134) = 2nd bicentric of X(291)
P(134) = trilinear pole of line X(6)P(6)

#### P(135) = BCA PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD and ANTICOMPLEMENTARY

Trilinears    (a^8-(b^2+c^2)*a^6+b^2*(2*b^2-c^2)*a^4-(b^2-c^2)*a^2*(b^4-2*b^2*c^2-c^4)-(b^2-c^2)*(b^6+2*b^4*c^2+c^6))/a : :
Barycentrics    (a^8-(b^2+c^2)*a^6+b^2*(2*b^2-c^2)*a^4-(b^2-c^2)*a^2*(b^4-2*b^2*c^2-c^4)-(b^2-c^2)*(b^6+2*b^4*c^2+c^6)) : :

P(135) and U(135) lie on the central line X(98)X(1297).
P(135) lies on non-central line P(38)P(136).

P(135) = 1st isobaryc of X(147)

#### P(136) = BCA-PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD and MEDIAL

Trilinears    (a^4-(b^2-c^2)*a^2-b^4-b^2*c^2+c^4)*(-b^4+a^2*c^2)/a : :
Barycentrics    (a^4-(b^2-c^2)*a^2-b^4-b^2*c^2+c^4)*(-b^4+a^2*c^2) : :

P(136) and U(136) lie on the central line X(9478)X(9479).
P(136) lies on these non-central lines: X(733)P(133), P(38)P(135), U(40)U(133)

P(136) = 1st isobaryc of X(8290)

#### P(137) = BCA-PERSPECTOR OF THESE TRIANGLES: ANTICOMPLEMENTARY and 1st BROCARD

Trilinears    (a^4+(b^2+c^2)*a^2+b^4+b^2*c^2-c^4)/a : :
Barycentrics    (a^4+(b^2+c^2)*a^2+b^4+b^2*c^2-c^4) : :

P(137) and U(137) lie on the central line X(83)X(826).
P(137) lies on these non-central lines: X(2)U(167), U(38)U(135), U(45)P(181). X(2896)P(167) U(38)U(135). (Randy Hutson, July 31, 2018)

P(137) = 2nd isobaryc of X(2896)

#### P(138) = BCA-PERSPECTOR OF THESE TRIANGLES: ANTI-McCAY and 1st PARRY

Trilinears    (a^6+3*(5*b^2-6*c^2)*a^4-3*(3*b^4-2*b^2*c^2-2*c^4)*a^2+(b^2-2*c^2)*(2*b^2-c^2)^2)/a : :
Barycentrics    (a^6+3*(5*b^2-6*c^2)*a^4-3*(3*b^4-2*b^2*c^2-2*c^4)*a^2+(b^2-2*c^2)*(2*b^2-c^2)^2) : :

P(138) and U(138) lie on the central line X(2)X(2793).

#### P(139) = BCA-PERSPECTOR OF THESE TRIANGLES: 3rd BROCARD and CIRCUMSYMMEDIAL

Trilinears    a*(2*(b^4+2*b^2*c^2-2*c^4)*a^4+b^2*c^2*(4*c^2+b^2)*a^2+2*b^4*c^4) : :
Barycentrics    a^2 *(2*(b^4+2*b^2*c^2-2*c^4)*a^4+b^2*c^2*(4*c^2+b^2)*a^2+2*b^4*c^4) : :

P(139) and U(139) lie on central line X(9488)X(9489).

#### P(140) = BCA-PERSPECTOR OF THESE TRIANGLES: 3rd BROCARD and SYMMEDIAL

Trilinears    a*(b^2*c^2+(b^2-c^2)*a^2)/b^2 : :
Barycentrics    a^2 *(b^2*c^2+(b^2-c^2)*a^2)/b^2 : :

P(140) and U(140) lie on central line X(9490)X(9491).

#### P(141) = BCA-PERSPECTOR OF THESE TRIANGLES: 3rd BROCARD and TANGENTIAL

Trilinears    a*((b^4+b^2*c^2-c^4)*a^4+b^2*c^2*(b^2+c^2)*a^2+b^4*c^4)
Barycentrics    a^2 *((b^4+b^2*c^2-c^4)*a^4+b^2*c^2*(b^2+c^2)*a^2+b^4*c^4)

P(141) and U(141) lie on central line X(83)X(9494).
P(141) lies on these non-central lines: X(6)U(159), X(7770)P(159)

#### P(142) = BCA-PERSPECTOR OF THESE TRIANGLES: EXCENTRAL and 2nd SHARYGIN

Trilinears    a^4-(b+c)*a^3-c*(b-2*c)*a^2-(b-c)*(b^2+2*b*c-c^2)*a+(b-c)*(b^3+2*b*c^2+c^3) : :
Barycentrics    a(a^4-(b+c)*a^3-c*(b-2*c)*a^2-(b-c)*(b^2+2*b*c-c^2)*a+(b-c)*(b^3+2*b*c^2+c^3)) : :

P(142) and U(142) lie on the central line X(103)X(105).
P(142) lies on these non-central lines: U(60)P(144), U(94)P(143)

#### P(143) = BCA-PERSPECTOR OF THESE TRIANGLES: INCENTRAL and 2nd SHARYGIN

Trilinears    (a^2+a*(b-c)+b^2-b*c-c^2)*(a*b-c^2) : :
Barycentrics    a*(a^2+a*(b-c)+b^2-b*c-c^2)*(a*b-c^2) : :

P(143) and U(143) lie on the central line X(4155)X(9507).
P(143) lies on these non-central lines: U(32)U(34), U(94)P(142)

#### P(144) = BCA-PERSPECTOR OF THESE TRIANGLES: 2nd MIXTILINEAR and 2nd SHARYGIN

Trilinears    a^7*c-(b^2+4*c^2)*a^6+(4*b^3+6*c^3-(2*b+c)*b*c)*a^5-(6*b^4+4*c^4-(b^2+2*b*c+4*c^2)*b*c)*a^4+(b-c)*(4*b^4-c^4+(5*b^2+11*b*c+5*c^2)*b*c)*a^3-b*(b-c)*(b^4+4*c^4+(3*b^2+b*c+7*c^2)*b*c)*a^2-b*c^2*(b-c)^4*a+b^3*c*(b-c)^4 : :

P(144) lies on these non-central lines: X(2)U(157), U(60)P(142).

#### P(145) = BCA-PERSPECTOR OF THESE TRIANGLES: 2nd PARRY and 3rd PARRY

Trilinears    (a^2-b^2)*b/c : :
Barycentrics    (a^2-b^2)/c^2 : :

P(145) and U(145) lie on the central line X(511)X(647).
P(145) lies on these non-central line: X(385)U(179), X(3060)P(157).

 Index tri prd bary prd bic sum bic diff crsum crdiff tri pole ideal pt midpt cevapt crspt vertcon 133 1966 385 5976 804 9467 9468 3978 804 5976 9469 5989 - 134 238 1914 8299 659 9470 292 239 812 9471 9472 8301 - 135 - 9473 98 2799 9474 9475 9476 2799 98 - - - 136 - 9477 9478 9479 - 9480 - - 9478 - - - 137 - 1031 83 826 9481 9482 9483 826 83 9484 - - 138 - - 2 9485 - 9486 9487 2793 - - - - 139 - - 9488 9489 - - - - - - - - 140 - - 9490 9491 9492 9493 - - - - - - 141 - - 83 9494 9495 9496 9497 9498 - - - - 142 9499 9500 105 2254 9501 9502 9503 2820 - - 9504 - 143 9505 9506 9507 9508 - 9509 9510 4155 - - - - 144 - - - 9511 - - - - - - - - 145 662 110 511 647 9512 1316 9513 511 647 - 9514 -

Bicentric pairs PU(146)-PU(159), together with prefatory notes and the table after PU(159), are contributed by César Lozada, March 28, 2016. The cevian triangles of the 1st and 2nd Brocard points are each perspective to each of these triangles:

ABC, at PU(1)
anticomplementary, at PU(45)
excentral, at PU(146)
Schroeter, at PU(147)
tangential, at PU(148)

The anticevian triangles of the 1st and 2nd Brocard points are each perspective to each of these triangles:

ABC, at PU(1)
extouch, at PU(149)
incentral, at PU(150)
intouch, at PU(151)
Lemoine, at PU(152)
Macbeath, at PU(153)
medial, at PU(154)
orthic, at PU(155)
Steiner, at PU(156)
symmedial, at PU(157)
Yff-contact, at PU(158)

The circumcevian triangles of the 1st and 2nd Brocard points are each perspective to each of these triangles:

ABC, at PU(1)
tangential, at PU(159)

#### P(146) = PERSPECTOR OF THE CEVIAN TRIANGLE OF P(1) AND THE EXCENTRAL TRIANGLE

Trilinears    a b^2 - b c^2 - c a^2 : :
Barycentrics    a (a b^2 - b c^2 - c a^2) : :

P(146) and U(146) lie on central line X(1019)X(7255).
P(146) lies on these non-central lines: X(1)U(1), X(9)U(160), X(1698)P(160). (Randy Hutson, July 31, 2018)

P(146) = P(1)-Ceva conjugate of X(1)

#### P(147) = PERSPECTOR OF THE CEVIAN TRIANGLE OF P(1) AND THE SCHROETER TRIANGLE

Trilinears    (a^4-(b^2+c^2)*a^2-b^4+b^2*c^2+c^4)*(b^2-c^2)/a : :
Barycentrics    (a^4-(b^2+c^2)*a^2-b^4+b^2*c^2+c^4)*(b^2-c^2) : :

P(147) and U(147) lie on central line X(2)X(4048).
P(147) = P(1)-Ceva conjugate of X(523)

#### P(148) = PERSPECTOR OF THE CEVIAN TRIANGLE OF P(1) AND THE TANGENTIAL TRIANGLE

Trilinears    a*(a^2*(b^2-c^2)-b^2*c^2) : :
Barycentrics    a^2 *(a^2*(b^2-c^2)-b^2*c^2) : :

P(148) and U(148) lie on central line X(2)X(669).
P(148) lies on these non-central lines: X(3)U(7), P(1)P(159)

P(148) = P(1)-Ceva conjugate of X(6)

#### P(149) = PERSPECTOR OF THE ANTICEVIAN TRIANGLE OF P(1) AND THE EXTOUCH TRIANGLE

Trilinears    (a^3*(a*b^2+a*c^2-2*c*b^2)-(b^4+c^4-(b+2*c)*b*c^2)*a^2-2*a*b^3*c^2+(b^2-c^2)*b^2*c^2)*c/b : :
Barycentrics    (a^3*(a*b^2+a*c^2-2*c*b^2)-(b^4+c^4-(b+2*c)*b*c^2)*a^2-2*a*b^3*c^2+(b^2-c^2)*b^2*c^2)/b^2 : :

P(149) lies on non-central line P(150)P(151).

P(149) = X(8)-Ceva conjugate of P(1)

#### P(150) = PERSPECTOR OF THE ANTICEVIAN TRIANGLE OF P(1) AND THE INCENTRAL TRIANGLE

Trilinears    c*(b*a^2-a*c^2+c*b^2)/b : :
Barycentrics    (b*a^2-a*c^2+c*b^2)/b^2 : :

P(150) lies on non-central line P(149)P(151).

P(150) and U(150) are collinear with X(17415).

P(150) = X(1)-Ceva conjugate of P(1)

#### P(151) = PERSPECTOR OF THE ANTICEVIAN TRIANGLE OF P(1) AND THE INTOUCH TRIANGLE

Trilinears    ((b^2+c^2)*a^3-(b^3+c^3-(b-c)*b*c)*a^2+a*b^2*c^2+b^2*c^2*(b-c))*c/b : :
Barycentrics    ((b^2+c^2)*a^3-(b^3+c^3-(b-c)*b*c)*a^2+a*b^2*c^2+b^2*c^2*(b-c))/b^2 : :

P(151) lies on non-central line P(149)P(150).

P(151) = X(7)-Ceva conjugate of P(1)

#### P(152) = PERSPECTOR OF THE ANTICEVIAN TRIANGLE OF P(1) AND THE LEMOINE TRIANGLE

Trilinears    ((2*b^2+c^2)*a^4-(b^4-2*b^2*c^2+2*c^4)*a^2+b^2*c^2*(2*b^2-c^2))*c/b : :
Barycentrics    ((2*b^2+c^2)*a^4-(b^4-2*b^2*c^2+2*c^4)*a^2+b^2*c^2*(2*b^2-c^2))/b^2 : :

P(152) lies on non-central line P(154)P(155).

P(152) = X(598)-Ceva conjugate of P(1)

#### P(153) = PERSPECTOR OF THE ANTICEVIAN TRIANGLE OF P(1) AND THE MacBEATH TRIANGLE

Trilinears    (a^6*c^2+(a^2-b^2)*a^2*(b^4-c^4-b^2*c^2)+(b^2-c^2)*b^2*c^4)*c/b : :
Barycentrics    (a^6*c^2+(a^2-b^2)*a^2*(b^4-c^4-b^2*c^2)+(b^2-c^2)*b^2*c^4)/b^2 : :

P(153) lies on non-central line X(3124)P(157).

P(153) = X(264)-Ceva conjugate of P(1)

#### P(154) = PERSPECTOR OF THE ANTICEVIAN TRIANGLE OF P(1) AND THE MEDIAL TRIANGLE

Trilinears    (b^2*c^2+a^2*(b^2-c^2))*c/b : :
Barycentrics    (b^2*c^2+a^2*(b^2-c^2))/b^2 : :

P(154) lies on these non-central lines: X(2086)P(157), P(152)P(155)
P(154) and U(154) lie on the central line X(3221)X(6375).

P(154) = X(2)-Ceva conjugate of P(1) =

#### P(155) = PERSPECTOR OF THE ANTICEVIAN TRIANGLE OF P(1) AND THE ORTHIC TRIANGLE

Trilinears    ((b^2+c^2)*a^4+(b^2*c^2-b^4-c^4)*a^2+(b^2-c^2)*b^2*c^2)*c/b : :
Barycentrics    ((b^2+c^2)*a^4+(b^2*c^2-b^4-c^4)*a^2+(b^2-c^2)*b^2*c^2)/b^2

P(155) lies on these non-central lines: U(1)P(157), P(152)P(154)

P(155) = X(4)-Ceva conjugate of P(1)

#### P(156) = PERSPECTOR OF THE ANTICEVIAN TRIANGLE OF P(1) AND THE STEINER TRIANGLE

Trilinears    (b^2*(a^4+b^2*c^2)-(b^2+c^2)*c^2*a^2)*c/b : :
Barycentrics    (b^2*(a^4+b^2*c^2)-(b^2+c^2)*c^2*a^2)/b^2

P(156) lies on these non-central lines: X(384)P(1), X(2076)P(2), X(3118)U(1) U(145)P(158). (Randy Hutson, July 31, 2018)

P(156) and U(156) are collinear with X(2531).

P(156) = X(99)-Ceva conjugate of P(1)

#### P(157) = PERSPECTOR OF THE ANTICEVIAN TRIANGLE OF P(1) AND THE SYMMEDIAL TRIANGLE

Trilinears    c*(a^2+b^2-c^2)/b : :
Barycentrics    (a^2+b^2-c^2)/b^2 : :

P(157) and U(157) lie on the central line X(51)X(647).
P(157) lies on these non-central lines: X(2)U(145), X(4)U(17), X(389)P(17), X(1316)P(1), X(2086)P(154), X(3060)P(145), X(3124)P(153), U(1)P(155)

P(157) = X(6)-Ceva conjugate of P(1)
P(157) = polar conjugate of P(38)

#### P(158) = PERSPECTOR OF THE ANTICEVIAN TRIANGLE OF P(1) AND THE YFF-CONTACT TRIANGLE

Trilinears    (a^3*b^2-c*(b^2-b*c+c^2)*a^2-a*b^2*c^2+b^3*c^2)*c/b : :
Barycentrics    (a^3*b^2-c*(b^2-b*c+c^2)*a^2-a*b^2*c^2+b^3*c^2)/b^2 : :

P(158) = X(190)-Ceva conjugate of P(1)

#### P(159) = PERSPECTOR OF THE CIRCUMCEVIAN TRIANGLE OF P(1) AND THE TANGENTIAL TRIANGLE

Trilinears    ((b^4-c^4)*a^4-b^4*c^4)*a : :
Barycentrics    (b^4-c^4)*a^4-b^4*c^4) : :

P(159) lies on central line X(76)X(9494).
P(159) lies on these non-central lines: X(6)U(141), X(7770)P(141), P(1)P(148)

 Index tri prd bary prd bic sum bic diff crsum crdiff tri pole ideal pt midpt cevapt crspt vertcon 146 1019 147 2 5113 148 2 669 3229 3225 149 150 151 152 153 154 3223 3224 6375 3221 3221 6375 155 156 2531 157 19 25 51 647 401 1987 436 158 159 76 9494

#### P(160) = SIMILICENTER OF 1ST MONTESDEOCA BISECTOR TRIANGLE AND EXCENTRAL TRIANGLE

Trilinears    a(-a2c + ab(b + c) + bc(2b + c) ): :
Barycentrics    a2(-a2c + ab(b + c) + bc(2b + c)) : :

In the plane of a triangle ABC, Let Ab be the point in which the internal bisector of angle A intersects the perpendicular bisector of segment AB, and define Bc and Ca cyclically. The triangle AbBcCa, here named the 1st Montesdeoca bisector triangle, is inversely similar to the excentral triangle, A'B'C', with similicenter P(160).

Define AcBaCb symmetrically. This 2nd Montesdeoca bisector triangle is also inversely similar to the excentral triangle, with similicenter U(160). The bicentric difference of PU(160) is X(1019), the center of the 1st Evans circle.

Let s1 be the similarity mappping from A'B'C' to AbBcCa, and let s2 be the similarity mapping from A'B'C' to AcBaCb. There is a unique point X for which s1(X) = s2(X), specifically, X = X(5540), and s1(X(5540)) = X(1083).

Contributed by Angel Montesdeoca, April 25, 2017. See HGT2017 and AdvGeom3769.

P(160) and U(160) lie on the central line X(1019)X(6626).

#### P(161) = ROTATION OF X(4) ABOUT X(3) BY π/2

Trilinears    a^2*(-a^2+b^2+c^2)-2*(b^2-c^2)*S : :
Trilinears    cos A - sin(B - C) : :
Barycentrics    a*(a^2*(-a^2+b^2+c^2)-2*(b^2-c^2)*S) : :

In the plane of a triangle ABC, rotation of U about V by π/2 means rotation in the "direction" of the shorter arc from A to B on the circumcircle of ABC (i.e., counterclockwise if ABC are labeled in counterclockwise order). Here, P(161) is the rotation of X(4) about X(3) by π/2, and it follows that U(161) is the rotation of X(4) about X(3) by -π/2.

The bicentric pairs PU(161) and PU(162), together with its vertices X(2) and X(4), lie on the Yff hyperbola, defined at Wolfram Mathworld. The foci of this hyperbola are X(3) and X(3830). A barycentric equation for the Yff hyperbola follows:

Σ( (SB+SC)*(SA-SB)*(SA-SC)*SA*x^2-2*S^2*(2*SW^2+(-7*R^2-2*SA)*SW-2*S^2-2*SA^2+15*R^2*SA)*y*z ) = 0.

(Contributed by César Lozada, August 21, 2017)

P(161) and U(161) lie on the Yff hyperbola and the central line {X(3), X(523).

P(161) lies on these non-central lines: {X(30),U(162)}, {X(381),P(162)}.
P(161) lies on these non-central lines: X(20)U(163), X(140)U(166), X(371)U(45), X(372)P(45), X(376)U(164), X(487)U(132), X(488)P(132), X(631)U(165), X(1656)P(166), X(2041)P(173), X(2042)U(173), X(3090)P(165), X(3091)P(163), X(3545)P(164), X(9131)U(63), X(9979)P(63) (Randy Hutson, July 31, 2018)

P(161) = reflection of M in N for these (M,N): (U(161), X(3)), (P(162), X(381))
P(161) = antipode of P(162) in Yff hyperbola

U(161) lies on the non-central lines: {X(30),P(162)}, {X(381),U(162)}.
X(20)P(163), X(140)P(166), X(371)P(45), X(372)U(45), X(376)P(164), X(487)P(132), X(488)U(132), X(631)P(165), X(1656)U(166), X(2041)U(173), X(2042)P(173), X(3090)U(165), X(3091)U(163), X(3545)U(164), X(9131)P(63), X(9979)U(63) (Randy Hutson, July 31, 2018)

U(161) = reflection of M in N for these (M,N): (P(161), X(3)), (U(162), X(381))
U(161) = antipode of U(162) in Yff hyperbola

#### P(162) = REFLECTION OF P(161) in X(381)

Trilinears    bc(5*a^4-(b^2+c^2)*a^2-2*(b^2-c^2)*(2*b^2-2*c^2-3*S)) : :
Barycentrics    5*a^4-(b^2+c^2)*a^2-2*(b^2-c^2)*(2*b^2-2*c^2-3*S) : :

P(162) and U(162) lie on the Yff hyperbola and on the central line {X(523), X(3830)}

P(162) lies on these non-central lines: {30,U(161)}, {381,P(161)}.
P(162) = reflection of M in N for these (M,N): (P(161), X(381)), (U(162), X(3830))
P(162) = antipode of P(161) in Yff hyperbola

U(162) lies on these non-central lines: {X(30),P(161)}, {X(381),U(161)}.
U(162) = reflection of M in N for these (M,N): (U(161), X(381)), (P(162), X(3830))
U(162) = antipode of U(161) in Yff hyperbola

#### P(163) = MOSES-YFF POINT H(3)

Trilinears    bc(-3*S^2+2*a^2*(b^2+c^2-a^2)-2*(b^2-c^2)*S) : :
Barycentrics    -3*S^2+2*a^2*(b^2+c^2-a^2)-2*(b^2-c^2)*S : :

Peter Moses found that if k is a function of a,b,c that is homogeneous in a,b,c of degree 0 (such as a constant), the points H(k) and H(-k) given by barycentrics

H(k) = (k^2-36)*S^2+6*(3*a^2*(-a^2+b^2+c^2) - (b^2-c^2)*k*S) : :

comprise a bicentric pair, and both points lie on the Yff hyperbola. The pair are here named the Moses-Yff-points; e.g., for k = 3, the points are the P(163) = Moses-Yff point H(3) and U(163) = Moses-Yff point H(-3).

P(163) and U(163) lie on Yff hyperbola and on the central line {X(523), X(3146)}

P(163) lies on these non-central lines: {X(3), U(165)}, {X(5), P(165)}, {X(20), U(161)}, {X(30), U(164)}, {X(381), P(164)}, {X(3091), P(161)}, {X(3523), U(166)}, {X(3543), U(162)}, {X(3839), P(162)}, {X(5056), P(166)}.
P(163) = reflection of M in N for these (M,N): (X(3146), U(163)), (X(381), P(164))
P(163) = antipode of P(164) in Yff hyperbola

U(163) lies on these non-central lines:
{X(3), P(165)}, {X(5), U(165)}, {X(20), P(161)}, {X(30), P(164)}, {X(381), U(164)}, {X(3091), U(161)}, {X(3523), P(166)}, {X(3543), P(162)}, {X(3839), U(162)}, {X(5056), U(166)}

U(163) = reflection of M in N for these (M,N): (P(163), X(3146)), (U(164), X(381))
U(163) = antipode of U(164) in Yff hyperbola

#### P(164) = MOSES-YFF POINT H(4)

Trilinears    bc(-10*S^2+9*a^2*(b^2+c^2-a^2) - 12*(b^2-c^2)*S) : :
Barycentrics    -10*S^2+9*a^2*(b^2+c^2-a^2) - 12*(b^2-c^2)*S : :

P(164) and U(164) lie on Yff hyperbola and on central line {523, 11001}

P(164) lies on these non-central lines: {X(30), U(163)}, {X(376), U(161)}, {X(381), P(163)}, {X(631), U(166)}, {X(3090), P(166)}, {X(3524), U(165)}, {X(3545), P(161)}, {X(5071), P(165)}.
P(164) = reflection of M in N for these (M,N): {X(381), P(163)}, {X(11001), U(164)}
P(164) = antipode of P(163) in Yff hyperbola

U(164) lies on the non-central lines: {X(30), P(163)}, {X(376), P(161)}, {X(381), U(163)}, {X(631), P(166)}, {X(3090), U(166)}, {X(3524), P(165)}, {X(3545), U(161)}, {X(5071), U(165)}.
U(164) = reflection of M in N for these (M,N): {X(381), U(163)}, {X(11001), P(164)}
U(164) = antipode of U(163) in Yff hyperbola

#### P(165) = MOSES-YFF POINT H(12)

Trilinears    bc(6*S^2+a^2*(b^2+c^2-a^2) - 4*(b^2-c^2)*S) : :
Barycentrics    6*S^2+a^2*(b^2+c^2-a^2) - 4*(b^2-c^2)*S : :

P(165) and U(165) lie on Yff hyperbola and on the central line {523, 3525}

P(165) lies on these non-central lines: {X(3), U(163)}, {X(5), P(163)}, {X(376), U(162)}, {X(631), U(161)}, {X(3090), P(161)}, {X(3524), U(164)}, {X(3533), U(166)}, {X(3545), P(162)}, {X(5071), P(164)}
P(165) = reflection of U(165) in X(3525)

U(165) lies on these non-central lines: {X(3), P(163)}, {X(5), U(163)}, {X(376), P(162)}, {X(631), P(161)}, {X(3090), U(161)}, {X(3524), P(164)}, {X(3533), P(166)}, {X(3545), U(162)}, {X(5071), U(164)}
U(165) = reflection of P(165) in X(3525)

#### P(166) = MOSES-YFF POINT H(18)

Trilinears    bc(16*S^2+a^2*(b^2+c^2-a^2) - 6*(b^2-c^2)*S) : :
Barycentrics    16*S^2+a^2*(b^2+c^2-a^2) - 6*(b^2-c^2)*S : :

P(166) and U(166) lie on the Yff hyperbola

P(166) lies on these non-central lines: {X(3), U(162)}, {X(5), P(162)}, {X(140), U(161)}, {X(523), U(166)}, {X(631), U(164)}, {X(1656), P(161)}, {X(3090), P(164)}, {X(3523), U(163)}, {X(3533), U(165)}, {X(5056), P(163)}

U(166) lies on these non-central lines: {X(3), P(162)}, {X(5), U(162)}, {X(140), P(161)}, {X(523), P(166)}, {X(631), P(164)}, {X(1656), U(161)}, {X(3090), U(164)}, {X(3523), P(163)}, {X(3533), P(165)}, {X(5056), U(163)}
The ideal point of PU(166) is X(523).

#### P(167) = 1st ISOBARYC OF X(83)

Trilinears    bc/(c2 + a2) : :
Barycentrics    1/(c2 + a2) : :

P(167) and U(167) lie on central line X(826)X(6292).
P(167) lies on these non-central lines: X(2)U(137), X(2896)P(137) P(1)P(11) U(40)U(133) (Randy Hutson, July 31, 2018)

The midpoint of PU(167) is X(6292).

#### P(168) =  1st ISOBARYC OF X(74)

Barycentrics    b^2 / (2 b^4 - (c^2 - a^2)^2 - b^2 (c^2 + a^2)) : :

Contributed by Randy Hutson, August 13, 2018

P(168) lies on these non-central lines: P(37)P(172), P(38)P(171), U(40)P(45), P(87)P(133)
P(168) = reflection of P(171) in P(38)
P(168) = trilinear quotient P(10)/P(86)
The midpoint of PU(168) is X(113).

#### P(169) =  1st ISOBARYC OF X(98)

Barycentrics    1/(c^4 + a^4 - b^2 c^2 - b^2 a^2) : :

Contributed by Randy Hutson, August 13, 2018

P(169) and U(169) lie on the central line X(114)X(132).
P(169) lies on these non-central lines: X(2)P(135), X(147)U(135), P(1)P(38), P(89)P(133), U(133)P(177)
P(169) = reflection of U(40) in P(38)
P(169) = trilinear quotient P(10)/P(88)
The midpoint of PU(169) is X(114).

#### P(170) = 1st ISOBARYC OF X(22)

Trilinears    bc(b2(c4 + a4 - b4)) : :
Barycentrics    b2(c4 + a4 - b4) : :

P(170) and U(170) lie on central line X(427)X(525).
P(170) lies on these non-central lines: X(2)P(37), X(1370)U(37), X(5133)U(38), X(7378)U(131). (Randy Hutson, July 31, 2018)

The midpoint of PU(170) is X(427).

#### P(171) = 1st ISOBARYC OF X(110)

Trilinears    bc(b2/(c2 - a2)) : :
Barycentrics    b2/(c2 - a2) : :

P(171) and U(171) lie on central line X(125)X(2799).
P(171) lies on these non-central lines: X(2)P(135), P(28)P168), X(148)U(40). (Randy Hutson, July 31, 2018)

The midpoint of PU(171) is X(125).

#### P(172) = 1st ISOBARYC OF X(125)

Trilinears    (csc A)(sin 2B)[sin(C - A)]2 : :
Barycentrics    (sin 2B)[sin(C - A)]2 : :

P(172) and U(172) lie on central line X(2799)X(5972).
P(172) lies on these non-central lines: X(2)P(135), P(37)P(168), X(99)P(40), X(110)U(171). (Randy Hutson, July 31, 2018)

#### P(173) = ORTHOCENTER OF 1st EHRMANN CIRCUMSCRIBING TRIANGLE

Trilinears    cos(B - C + π/3) : :

See preamble before X(18300).

P(173) and U(173) lie on the Hatzipolakis axis (line X(5)X(523)), as do the points PU(5). The midpoint of PU(173) is X(5).

P(173) lies on these non-central lines:
X(49)X, where X = isogonal conjugate of U(5)
X(54)X, where X = isogonal conjugate of P(5)
X(94)X, where X = P(5)-Ceva conjugate of U(5)
X(110)X, where X = isogonal conjugate of U(173)
X(635)P(11)
X(636)U(11)
X(18351)X, where X = U(173)-Ceva conjugate of P(173)

P(173) = X(4)-of-1st-Ehrmann-circumscribing-triangle
P(173) = X(3)-of-1st-Ehrmann-inscribed-triangle
P(173) = orthologic center of these triangles: 1st Ehrmann circumscribing to 2nd Ehrmann inscribed

#### P(174) =  1st EHRMANN PIVOT OF 2nd EHRMANN CIRCUMSCRIBING TRIANGLE

Barycentrics    3 a^4 - 3 b^4 - 3 c^4 + 6 b^2 c^2 + 2 Sqrt (b^2 - c^2) S : :

Contributed by Randy Hutson, August 13, 2018

U(174) is the 2nd Ehrmann pivot of 1st Ehrmann circumscribing triangle.
P(5) is the 1st Ehrmann pivot of 1st Ehrmann circumscribing triangle.
U(5) is the 2nd Ehrmann pivot of 2nd Ehrmann circumscribing triangle.

P(174) and U(174) lie on the central line X(4)X(523).
P(174) lies on these non-central lines: X(3)U(5), X(5)U(175), X(30)P(175), X(381)P(5), X(382)U(173), X(3843)P(173), X(5318)P(45), X(5321)U(45)

The midpoint of PU(174) is X(4).

#### P(175) =  1st EHRMANN PIVOT OF 2nd EHRMANN INSCRIBED TRIANGLE

Barycentrics    3 a^4 - 3 a^2 b^2 - 3 a^2 c^2 - 2 Sqrt (b^2 - c^2) S : :

Contributed by Randy Hutson, August 13, 2018

U(175) is the 2nd Ehrmann pivot of 1st Ehrmann inscribed triangle.
P(5) is the 1st Ehrmann pivot of 1st Ehrmann inscribed triangle.
U(5) is the 2nd Ehrmann pivot of 2nd Ehrmann inscribed triangle.

P(175) and U(175) lie on the central line X(3)X(523).
P(175) lies on these non-central lines: X(2)P(5), X(4)U(5), X(5)U(174), X(15)P(45), X(16)U(45), X(20)U(173), X(30)P(174), X(631)P(173)

The midpoint of PU(175) is X(3).
P(175) = anticomplement of P(5)

#### P(176) =  1st ISOBARYC OF X(30)

Barycentrics    2 b^4 - (c^2 - a^2)^2 - b^2 (c^2 + a^2) : :

Contributed by Randy Hutson, August 13, 2018

P(176) and U(176) lie on the line at infinity.
P(176) lies on these non-central lines: X(2)P(37), X(376)U(37), U(43)P(186)

P(176) = isogonal conjugate of P(87)

#### P(177) =  1st ISOBARYC OF X(511)

Barycentrics    sin B cos(B + ω) : :
Barycentrics    b^2 (b^2 c^2 + b^2 a^2 - c^4 - a^4) : :

Contributed by Randy Hutson, August 13, 2018

P(177) and U(177) lie on the line at infinity.
P(177) lies on these non-central lines: P(1)P(37), P(11)P(38), U(37)U(189), U(45)U(191), U(133)P(169)

P(177) = isogonal conjugate of P(89)

#### P(178) =  1st ISOBARYC OF X(512)

Barycentrics    sin^2 B (cos 2C - cos 2A) : :
Barycentrics    b^2 (c^2 - a^2) : :

Contributed by Randy Hutson, August 13, 2018

P(178) and U(178) lie on the line at infinity.
P(178) lies on these non-central lines: X(76)U(1), X(99)U(2), X(194)P(1), X(384)U(183), X(385)P(2), X(7760)P(183), X(9983)U(182), X(11257)P(189), X(12251)U(189), X(18829)P(40), X(18906)U(191), U(17)P(37)

P(178) = isogonal conjugate of P(91)

#### P(179) =  1st ISOBARYC OF X(523)

Barycentrics    c^2 - a^2 : :

Contributed by Randy Hutson, August 13, 2018

P(179) and U(179) lie on the line at infinity.
P(179) lies on these non-central lines: X(2)U(43), X(6)U(11), X(69)P(11), X(193)U(132), X(385)U(145), X(892)P(40), X(14999)P(172), X(20080)P(132), U(181)U(191)

P(179) = isogonal conjugate of P(105)

#### P(180) =  1st ISOBARYC OF X(524)

Barycentrics    2 cot B - cot C - cot A : :
Barycentrics    2 b^2 - c^2 - a^2 : :

Contributed by Randy Hutson, August 13, 2018

P(180) and U(180) lie on the line at infinity.
P(180) lies on these non-central lines: X(2)P(11), X(599)U(11), X(11160)U(132), X(18823)P(40)

P(180) = isogonal conjugate of P(107)

#### P(181) =  1st BROCARD POINT OF 1st BROCARD TRIANGLE

Barycentrics    c^2 (a^4 + b^4 + a^2 b^2) : :

Contributed by Randy Hutson, August 13, 2018

P(181) lies on these non-central lines: X(2)U(105), U(1)U(154), U(11)U(38), U(45)P(137), U(179)P(191)

The ideal point of PU(181) is X(804). The midpoint of PU(181) is X(24256).

P(181) = 2nd isobaryc of X(3094)
P(181) and U(181) lie on central line X(804)X(24256)

#### P(182) =  1st BROCARD POINT OF 5th BROCARD TRIANGLE

Barycentrics    a^2 (a^4 b^2 - b^4 c^2 - a^2 c^4 - b^2 c^4 - c^6) : :

Contributed by Randy Hutson, August 13, 2018

P(182) lies on these non-central lines: X(32)P(1), X(2076)P(39), X(3094)U(1), X(3098)U(189), X(9821)P(189), X(9983)U(178), U(167)U(181)

The ideal point of PU(182) is X(512).
P(182) = P(1)-of-circumcevian-triangle-of-P(1)

#### P(183) =  1st BROCARD POINT OF 5th ANTI-BROCARD TRIANGLE

Barycentrics    a^2 (a^2 + c^2) : :

Contributed by Randy Hutson, August 13, 2018

P(183) and U(183) lie on the central line X(512)X(5007).
P(183) lies on these non-central lines: X(6)U(1), X(32)P(1), X(39)P(2), X(182)U(189), X(384)U(178), X(5039)U(191), X(7760)P(178)

The midpoint of PU(183) is X(5007).

#### P(184) =  1st BROCARD POINT OF 6th BROCARD TRIANGLE

Barycentrics    2 a^10 b^2 c^2 + a^2 b^2 c^8 (b^2 - 2 c^2) + c^8 (b^3 - b c^2)^2 + a^6 (b^2 + 2 c^2) (2 b^6 + 2 b^2 c^4 - c^6) - a^8 (b^6 + 4 b^4 c^2 + 4 b^2 c^4 - c^6) - a^4 (b^10 + b^8 c^2 + 4 b^6 c^4 - 4 b^4 c^6 + b^2 c^8 - c^10) : :

Contributed by Randy Hutson, August 13, 2018

P(184) lies on these non-central lines: X(3)P(133), X(76)U(133), X(384)P(181)

The ideal point of PU(184) is X(804).

#### P(185) =  1st BROCARD POINT OF 6th ANTI-BROCARD TRIANGLE

Barycentrics    (a^2 - b^2) (a^4 - b^2 c^2) (a^2 b^4 + a^4 c^2 - 3 a^2 b^2 c^2 + b^2 c^4) : :

Contributed by Randy Hutson, August 13, 2018

P(185) and U(185) lie on the central line X(804)X(4107).
P(185) lies on these non-central lines: X(99)P(91), X(182)P(181), X(4027)P(133), X(4048)U(181), X(4577)U(40), X(4590)P(2), X(8290)U(133)

The midpoint of PU(185) is X(5026).

#### P(186) =  1st BROCARD POINT OF ORTHOCENTROIDAL TRIANGLE

Barycentrics    a^4 + 2 a^2 b^2 - b^4 + b^2 c^2 : :

Contributed by Randy Hutson, August 13, 2018

P(186) and U(186) lie on the central line X(690)X(7753).
P(186) lies on these non-central lines: X(6)U(40), X(1316)U(1), X(5475)P(40), P(1)P(43), U(43)P(176)

The midpoint of PU(186) is X(7753).

#### P(187) =  1st BROCARD POINT OF ANTI-ORTHOCENTROIDAL TRIANGLE

Barycentrics    a^2 (a^6 (b^2 + c^2) - 2 a^4 (b^4 + c^4) + a^2 (b^6 + 3 b^4 c^2 - 3 b^2 c^4 + c^6) - 2 b^6 c^2 + b^4 c^4 + b^2 c^6) : :

Contributed by Randy Hutson, August 13, 2018

P(187) and U(187) lie on the central line X(690)X(1569).
P(187) lies on these non-central lines: X(6)U(40), X(74)P(89), X(3016)P(40)

#### P(188) =  1st BROCARD POINT OF PEDAL TRIANGLE OF 2nd BROCARD POINT

Barycentrics    a^2 (a^4 c^2 + a^2 c^4 + 3 a^2 b^2 c^2 + a^2 b^4 + b^4 c^2 - b^6) : :

Contributed by Randy Hutson, August 13, 2018

U(188) is the 2nd Brocard point of the pedal triangle of the 1st Brocard point.
P(1) is the 1st Brocard point of the pedal triangle of the 1st Brocard point.
U(1) is the 2nd Brocard point of the pedal triangle of the 2nd Brocard point.

P(188) lies on these non-central lines: X(6)P(1) X(3095)U(1) X(3398)U(2)

The ideal point of PU(188) is X(512).

#### P(189) =  1st BROCARD POINT OF ANTIPEDAL TRIANGLE OF 2nd BROCARD POINT

Barycentrics    a^2 (2 a^4 c^2 - 2 a^2 c^4 + a^4 b^2 - 3 b^2 c^4 - b^6) : :

Contributed by Randy Hutson, August 13, 2018

U(189) is the 2nd Brocard point of the antipedal triangle of the 1st Brocard point.
P(1) is the 1st Brocard point of the antipedal triangle of the 1st Brocard point.
U(1) is the 2nd Brocard point of the antipedal triangle of the 2nd Brocard point.

P(189) and U(189) lie on the 2nd Brocard circle and the central line X(512)X(5188).
P(189) lies on these non-central lines: X(3)U(1), X(511)P(1), X(5171)P(2), X(9737)U(190), X(11257)P(178), X(12251)U(178), X(13354)U(191), P(37)U(177)

Let Ua, Ub, Uc be the reflections of U(1) in A, B, C, resp.; then P(189) is the circumcenter of UaUbUc.

The midpoint of PU(189) is X(5188).

P(189) = reflection of U(1) in X(3)
P(189) = antipode in 2nd Brocard circle of U(1)

#### P(190) =  1st BROCARD POINT OF REFLECTION TRIANGLE OF 2nd BROCARD POINT

Barycentrics    a^2 (a^4 b^2 + 2 a^2 b^2 c^2 - b^4 c^2 + a^2 c^4 + b^2 c^4 - c^6) : :

Contributed by Randy Hutson, August 13, 2018

U(190) is the 2nd Brocard point of the reflection triangle of the 1st Brocard point.
P(1) is the 1st Brocard point of the reflection triangle of the 1st Brocard point.
U(1) is the 2nd Brocard point of the reflection triangle of the 2nd Brocard point.

P(190) lies on these non-central lines: X(32)P(1), X(576)U(191), X(3095)U(1), X(9737)U(189), X(13330)P(191)

The ideal point of PU(190) is X(512).
P(190) = circle-{{X(4),X(194),X(3557),X(3558)}}-inverse of U(1)

#### P(191) =  HOMOTHETIC CENTER OF PEDAL TRIANGLE OF 1st BROCARD POINT AND ANTIPEDAL TRIANGLE OF 2nd BROCARD POINT

Barycentrics    a^2 (b^4 - a^2 b^2 - b^2 c^2 - 2 c^2 a^2) : :

Contributed by Randy Hutson, August 13, 2018

P(191) and U(191) lie on the central line X(512)X(5052).
P(191) lies on these non-central lines: X(6)U(1), X(182)P(2), X(511)P(1), X(576)U(190), X(5017)U(2), X(13330)P(190), X(13354)U(189), X(18906)U(178), P(45)U(177), P(89)P(105), U(179)P(181)

The midpoint of PU(191) is X(5052).

P(191) = reflection of U(1) in X(6)
P(191) = endo-homothetic center of the pedal triangle of 2nd Brocard point and antipedal triangle of 1st Brocard point
P(191) = U(191)-of-pedal-triangle-of-P(1)
P(191) = U(191)-of-antipedal-triangle-of-U(1)

#### P(192) =  HOMOTHETIC CENTER OF REFLECTION TRIANGLE OF 1st BROCARD POINT AND ANTIPEDAL TRIANGLE OF 2nd BROCARD POINT

Barycentrics    a^2 (2 a^4 c^2 + a^4 b^2 + 2 a^2 b^2 c^2 - b^2 c^4 - b^6) : :

Contributed by Randy Hutson, August 13, 2018

P(192) and U(192) lie on circle {{X(371),X(372),PU(1),PU(39)}}.
P(192) lies on these non-central lines: X(3)U(188), X(32)U(1), X(511)P(1), X(5017)U(39), X(13330)P(188)

The ideal point of PU(192) is X(512).

P(192) = reflection of U(1) in X(32)
P(192) = U(39)-of-reflection-triangle-of-P(1)
P(192) = U(39)-of-antipedal-triangle-of-U(1)

#### P(193) =  1ST REAL FOCUS OF THE ORTHIC INCONIC

Barycentrics    a^2*(4*(a^2 - b^2)*(a^2 - c^2)*(b^2 - c^2)*(-3 + J)*S^2 + (3*(a^2 - b^2)*(a^2 - c^2) + (a^2*b^2 - b^4 + a^2*c^2 - c^4)*J - b^2*c^2*J^2)*Sqrt[2*(a^4*b^4*c^4*J^3 - 8*S^6 - 27*SA^2*SB^2*SC^2 + 18*S^2*SA*SB*SC*SW + S^4*SW^2)]) : :
Trilinears    (2*(a*(SA^2-SB*SC)*OH+a*R*(S^2+3*SA*SW)-(b^2+c^2)*S*b*c))*Sqrt[2*R*(OH^3-9*R*(3*R^2-SW))-S^2-SW^2]-a*(SB-SC)*(SC-SA)*(SA-SB)*(OH-3*R) : :
(César Lozada, Dec. 8, 2021)

Contributed by Peter Moses, November 16, 2020

The 2nd real focus of the orthic inconic is U(193).

P(193) and U(193) lie on the curves K018, K019, K040, K048, K102, K164, K168, K188, K189, K190, K233, K314, K417, K418, K708, K754, K755, K756, K757, K874, K1093, Q002, Q003, Q021, Q056, Q112, Q149, and on this line: {6, 1345}

P(193) = reflection of U(193) in X(6)
P(193) = isogonal conjugate of U(193)
P(193) = orthic-isogonal conjugate of U(193)
P(193) = psi-transform of U(193)
P(193)) = X(4)-Ceva conjugate of U(193)
P(193) = X(3)-cross conjugate of U(193)
P(193) = {X(i),X(j)}-harmonic conjugate of X for these (i,j,X): {1345, 8106, U(193)}, {14899, 35609, U(193)}

U(193) = reflection of P(193) in X(6)
U(193) = isogonal conjugate of P(193)
U(193) = orthic-isogonal conjugate of P(193)
U(193) = psi-transform of P(193)
U(193)) = X(4)-Ceva conjugate of P(193)
U(193) = X(3)-cross conjugate of P(193)
U(193) = {X(i),X(j)}-harmonic conjugate of X for these (i,j,X): {1345, 8106, P(193)}, {14899, 35609, P(193)}

#### P(194) =  1ST IMAGINARY FOCUS OF THE ORTHIC INCONIC

Barycentrics    a^2*(4*(a^2 - b^2)*(a^2 - c^2)*(b^2 - c^2)*(3 + J)*S^2 - (3*(a^2 - b^2)*(a^2 - c^2) - (a^2*b^2 - b^4 + a^2*c^2 - c^4)*J - b^2*c^2*J^2)*Sqrt[2*(-(a^4*b^4*c^4*J^3) - 8*S^6 - 27*SA^2*SB^2*SC^2 + 18*S^2*SA*SB*SC*SW + S^4*SW^2)]) : :

Contributed by Peter Moses, November 16, 2020

The 2nd imaginary focus of the orthic inconic is U(194). See P(193) for the 1st and 2nd real foci.

P(194) and U(194) lie on the curves K018, K019, K040, K048, K102, K164, K168, K188, K189, K190, K233, K314, K417, K418, K708, K754, K755, K756, K757, K874, K1093, Q002, Q003, Q021, Q056, Q112, Q149, and on this line: {6, 1345}

P(194) = reflection of U(194) in X(6)
P(194) = isogonal conjugate of U(194)
P(194) = orthic-isogonal conjugate of U(194)
P(194) = psi-transform of U(194)
P(194)) = X(4)-Ceva conjugate of U(194)
P(194) = X(3)-cross conjugate of U(194)
P(194) = {X(i),X(j)}-harmonic conjugate of X for these (i,j,X): {1345, 8106, U(194)}, {14899, 35609, U(194)}

U(194) = reflection of P(194) in X(6)
U(194) = isogonal conjugate of P(194)
U(194) = orthic-isogonal conjugate of P(194)
U(194) = psi-transform of P(194)
U(194)) = X(4)-Ceva conjugate of P(194)
U(194) = X(3)-cross conjugate of P(194)
U(194) = {X(i),X(j)}-harmonic conjugate of X for these (i,j,X): {1345, 8106, P(194)}, {14899, 35609, P(194)}

#### P(195) =  1ST MOSES POINT

Barycentrics    a*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4 + 2*a^3*b^2*c - 2*a^2*b^3*c - a^3*b*c^2 + a*b^3*c^2 + b^4*c^2 - a^3*c^3 - a*b^2*c^3 - 2*b^3*c^3 + 2*a^2*c^4 + a*b*c^4 + b^2*c^4 - a*c^5) : :

Contributed by Dan Reznik, November 25, 2020

The 1st and 2nd Moses points are the incircle-inverses of the 1st and 2nd Brocard points, respectively. See X(39541) and X(40458)-X(40461) for related triangle centers.

P(195) = incircle-inverse of P(1)
U(195) = incircle-inverse of U(1)

#### P(196) =  1ST MONTESDEOCA POINT

Barycentrics    b^2 c^2 (b^2-c^2-3 a^2) : :

Contributed by Angel Montesdeoca, February 19, 2021

Let ℓab be the axis of the parabola tangent to AC at A and also tangent to BG at B, and define ℓbc and ℓca cyclically. Let
Ab = ℓbc∩ℓca
Bc = ℓca∩ℓab
Ca = ℓab∩ℓbc.

The triangle AbBcCa is homothetic to ABC, and the center of homothety is P(196).

Let ℓac be the axis of the parabola tangent to AB at A and also tangent to CG at C, and define ℓba and ℓcb cyclically. Let
Ac= ℓba∩ℓcb
Ba = ℓcb∩ℓac
Cb = ℓac∩ℓba.

The triangle AcBaCb is homothetic to ABC, and the center of homothety is the 2nd Montesdeoca point, U = b^2 c^2 (b^2-c^2+3 a^2) : :

The triangles AbBcCa and AcBaCb are homothetic, and their homothetic center is X(41300).

P and U are a bicentric pair that lie on the line X(2)X(647).
X(850) = bicentric sum of PU
X(2) = bicentric difference of PU
X(237) = crossdifference of PU
X(290) = trilinear pole of the line PU
X(23878) = ideal point of line PU

#### P(197) =  1ST REAL FOCUS OF THE DE LONGCHAMPS ELLIPSE

Barycentrics    a*(2*sqrt(a*b*c)+sqrt(a+b+c)*(b-c)) : :

Contributed by César E. Lozada, December 05, 2021.

The 2nd real focus of the de Longchamps ellipse is U(197)

P(197) and U(197) lie on the line: {1, 513}

#### P(198) =  1ST CIRCUMCIRCLE-INTERCEPT OF LINE X(3)X(523)

Barycentrics    a^2 (a^2-b^2-c^2) J - 2 (b^2-c^2) S : :

Contributed by Peter Moses, January 12, 2022.

The 2nd circumcircle-intercept of X(3)X(523) is

U(198) = a^2 (a^2-b^2-c^2) J + 2 (b^2-c^2) S : : ,

where is identified at X(1113) by the equation

J = |OH|/R = (1/abc)[S(6) - S(2,4) + 3a^2 b^2 c^2]^(1/2)

The line X(3)X(523) meets the Euler line at right angles at X(3), so that X(1113), X(1114), P(198), U(198) are the vertices of a square inscribed in the circumcircle.

X(32710) = vertex conjugate of PU(198)

#### P(199) =  1ST POINT ON NINE-POINT CIRCLE AT WHICH THE TANGENT MEETS THE EULER LINE IN X(1113)

Barycentrics    (a^2 - b^2 - c^2)*(a^6*b^2 - a^4*b^4 - a^2*b^6 + b^8 + a^6*c^2 + a^2*b^4*c^2 - 4*b^6*c^2 - a^4*c^4 + a^2*b^2*c^4 + 6*b^4*c^4 - a^2*c^6 - 4*b^2*c^6 + c^8) - 2*a^2*b^2*c^2*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*J + 2*a^2*b^2*c^2*(b^2 - c^2)*Sqrt[(-3 + J)*(-1 + J)]*S : :

Contributed by Peter Moses, January 14, 2022.

The 2nd such touchpoint, U(199), is given by

(a^2 - b^2 - c^2)*(a^6*b^2 - a^4*b^4 - a^2*b^6 + b^8 + a^6*c^2 + a^2*b^4*c^2 - 4*b^6*c^2 - a^4*c^4 + a^2*b^2*c^4 + 6*b^4*c^4 - a^2*c^6 - 4*b^2*c^6 + c^8) - 2*a^2*b^2*c^2*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*J - 2*a^2*b^2*c^2*(b^2 - c^2)*Sqrt[(-3 + J)*(-1 + J)]*S : : ,

in which the coefficient in the last term is -2, instead of +2 as in P(199).

The midpoint of PU(199) is X(46681).

#### P(200) =  1ST POINT ON NINE-POINT CIRCLE AT WHICH THE TANGENT MEETS THE EULER LINE IN X(1114)

Barycentrics    (a^2 - b^2 - c^2)*(a^6*b^2 - a^4*b^4 - a^2*b^6 + b^8 + a^6*c^2 + a^2*b^4*c^2 - 4*b^6*c^2 - a^4*c^4 + a^2*b^2*c^4 + 6*b^4*c^4 - a^2*c^6 - 4*b^2*c^6 + c^8) + 2*a^2*b^2*c^2*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*J + 2*a^2*b^2*c^2*(b^2 - c^2)*Sqrt[(-3 - J)*(-1 - J)]*S : :

Contributed by Peter Moses, January 14, 2022.

The 2nd such touchpoint, U(200), is given by

(a^2 - b^2 - c^2)*(a^6*b^2 - a^4*b^4 - a^2*b^6 + b^8 + a^6*c^2 + a^2*b^4*c^2 - 4*b^6*c^2 - a^4*c^4 + a^2*b^2*c^4 + 6*b^4*c^4 - a^2*c^6 - 4*b^2*c^6 + c^8) + 2*a^2*b^2*c^2*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*J - 2*a^2*b^2*c^2*(b^2 - c^2)*Sqrt[(-3 - J)*(-1 - J)]*S : : ,

in which the coefficient in the last term is -2, instead of +2 as in P(200).

The midpoint of PU(200) is X(46682).

#### P(201) =  POINT WITH EULER COORDINATES (0, SA^2+SB^2+SC^2)

Barycentrics    a^6 - a^4 b^2 - 2 a^2 b^4 + 2 a^2 b^2 c^2 - b^4 c^2 + c^6 : :

This point is used to define the positive y-axis (the orthic axis) in the Euler coordinate system.

The bicentric mate, U(201), has Euler coordinates (0, (a^2+b^2+c^2)/4); this is the reflection of P(201) in X(468), which is the origin (where the orthic axis meets the Euler line, with Euler coordinates (0,0)).

In Euler coordinates (0,y), for y = SA^2+SB^2+SC^2, we also have y = SW^2 = (SA+SB+SC)^ 2 = (1/4)(a^2+b^2+c^2^2.

The points P(201) and U(201) lie on the orthic axis, X(230)X(231). Their midpoint is X(468).

#### P(202) =  POINT WITH EULER COORDINATES (0, S^2)

Barycentrics    a^6 - 2 a^2 b^4 + b^6 - a^4 c^2 + 2 a^2 b^2 c^2 - 2 b^4 c^2 + b^2 c^4 : :

The points P(202) and U(202) lie on the orthic axis, X(230)X(231). Their midpoint is X(468).

#### P(203) =  POINT WITH EULER COORDINATES (0, SA^2 + SB^2 + SC^2)

Barycentrics    a^6 - 2 a^4 b^2 - b^6 + a^4 c^2 + 2 a^2 b^2 c^2 + 2 b^4 c^2 - 2 a^2 c^4 - 3 b^2 c^4 + 2 c^6 : :

The points P(203) and U(203) lie on the orthic axis, X(230)X(231). Their midpoint is X(468).

#### P(204) =  POINT WITH EULER COORDINATES (0, a^4 + b^4 + c^4)

Barycentrics    2 a^6 - 5 a^4 b^2 - 2 a^2 b^4 - 3 b^6 + 3 a^4 c^2 + 4 a^2 b^2 c^2 + 3 b^4 c^2 - 2 a^2 c^4 - 5 b^2 c^4 + 5 c^6 : :

The points P(204) and U(204) lie on the orthic axis, X(230)X(231). Their midpoint is X(468).

#### P(205) =  POINT WITH EULER COORDINATES (0, (a^2 + b^2 + c^2)^2

Barycentrics    2 a^6 - 5 a^4 b^2 - 10 a^2 b^4 - 3 b^6 + 3 a^4 c^2 + 4 a^2 b^2 c^2 - 5 b^4 c^2 + 6 a^2 c^4 + 3 b^2 c^4 + 5 c^6 : :

The points P(205) and U(205) lie on the line X(468)X(669).

#### P(206) =  POINT WITH EULER COORDINATES (0, (E - 8F)S^2/(E+F)

Barycentrics    2 a^8 - 7 a^6 b^2 + 5 a^4 b^4 + 7 a^2 b^6 - 7 b^8 + 9 a^6 c^2 + 2 a^4 b^2 c^2 - 31 a^2 b^4 c^2 + 16 b^6 c^2 - 11 a^4 c^4 + 33 a^2 b^2 c^4 - 2 b^4 c^4 - 9 a^2 c^6 - 16 b^2 c^6 + 9 c^8 : :

The points P(206) and U(206) lie on the line the orthic axis, X(230)X(231).

#### P(207) =  POINT WITH EULER COORDINATES (0, 3FS)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(3*(a^2 - b^2 - c^2)*(b^2 - c^2) + 2*(2*a^2 - b^2 - c^2)*S) : :

P(207) is obtained from by a glide-rotation of X(4) = (3F,0); specifically, rotate X(4) about X(468) through a right angle toward the positive y-axis, with dilation factor S.

The points P(207) and U(207) lie on the line the orthic axis, X(230)X(231).

#### P(208) =  POINT WITH EULER COORDINATES (0, R^2 S)

Barycentrics    a^2*b^2*c^2*(b^2 - c^2) - (2*a^2 - b^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*S : :

P(208) is obtained from by a glide-rotation of X(44452) = (R^2,0); specifically, rotate X(44452) about X(468) through a right angle toward the positive y-axis, with dilation factor S.

The points P(208) and U(208) lie on the line the orthic axis, X(230)X(231).

#### P(209) =  POINT WITH EULER COORDINATES (0, (3/2) R^2 S)

Barycentrics    3*a^2*b^2*c^2*(b^2 - c^2) - 2*(2*a^2 - b^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*S : :

P(209) is obtained from by a glide-rotation of X(140) = ((3/2)R^2,0); specifically, rotate X(140) about X(468) through a right angle toward the positive y-axis, with dilation factor S.

The points P(209) and U(209) lie on the line the orthic axis, X(230)X(231).

#### P(210) =  3RD MONTESDEOCA POINT

Barycentrics    b^2 (b^2-c^2) (a^2+b^2-c^2) (a^2-b^2+c^2)^2 : :

Contributed by Angel Montesdeoca, August 8, 2022

In the plane of a triangle ABC, let

A'B'C' = orthic triangle;
M = a variable point on line BC;
M1 = point on B'C' such that ∠MAM1 = ∠ A;
M2 = point on B'C' such that ∠MAM2 = - ∠ A;
p1 = line through M1 perpendicular to B'C';
p2 = line through M2 perpendicular to B'C';
h1 = line through M1 perpendicular to BC;
h2 = line through M2 perpendicular to BC;
H1 = p1 ∩ h1;
H2 = p2 ∩ h2;
1 = locus of H1 as M traverses BC;
2 = locus of H2 as M traverses BC;
A1 = center of ℋ1, and define B1 and C1 cyclically;
A2 = center of ℋ2, and define B2 and C2 cyclically;
σ1 = the affine transformation that maps ABC onto A1B1C1;
σ2 = the affine transformation that maps ABC onto A2B2C2;
F1 = unique finite fixed point of σ1;
F2 = unique finite fixed point of σ2.
Then F1 and F2 are a bicentric pair of points, and
P(210) = F1= b^2(b^2-c^2)(a^2+b^2-c^2)(a^2-b^2+c^2) : :
U(210) = F2 = c^2(b^2-c^2)(a^2+b^2-c^2)(a^2-b^2+c^2) : :

For a construction of P(210) and U(210), labeled as σ1(X) and σ2(X), see PU(210)

X(115) = bicentric sum of P(210) and U(210).

1st edition, PU(1) to PU(42): 9/22/2003
2nd edition, to PU(85): 8/22/2004
3rd edition, to PU(111): 9/01/2013
4th edition, to PU(119): 1/06/2015
5th edition, to PU(145): 1/29/2016
6th edition, to PU(159): 4/20/2016
7th edition, to PU(166): 9/01/2017
8th edition, to PU(192): 9/14/2018
9th edition, to PU(194): 11/18/2020
10th edition, to PU(197): 12/06/2021, by César Lozada.