Definition of bicentric pairs, 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 (inGlossary), but that|f(a,b,c)| is Thennotequal to |f(a,c,b)|.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 abicentric pair of points, or simply, abicentric pair.Examples include the Brocard points, given in trilinear coordinates byc/b : a/c : b/a and b/c : c/a : a/b. In the list below that starts with "P(1) = 1st BROCARD POINT", only thefirst trilinearof thefirst pointof the pair is given. These first points 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 thenaf(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 pairsP = p : q : randU = u : v : wonto triangle centers. For two of these operations, it is necessary that p, q, r and u,v,w be represented byf(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 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 a ^{2}(qw + rv) : b^{2}(ru + pw) : c^{2}(pv + qu)Crossdifference a ^{2}(qw - rv) : b^{2}(ru - pw) : c^{2}(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 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. 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 Geometricorum3 (2003) 35-47; Cubics Associated with Triangles of Equal Areas,Forum Geometricorum1 (2001) 161-171; and "Enumerative triangle geometry, part 1: the primary system,S,"Rocky Mountain Journal of Mathematics32 (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 X

_{Y}and X_{Z}defined byX are the_{Y}= y : z : x and X_{Z}= z : x : y1st and 2nd bicentrics of X, respectively. Following the method in[Areas], the2nd equal-areas cubicis introduced here (Oct. 5, 2003) asthe locus of a point X such that the areas of the cevian triangles of XAn equation for this cubic is_{Y}and X_{Z}are equal.(bz + cx)(ay + cx)(ay + bz) = (bx + cy)(az + cy)(az + bx), that is, the cubic Z(X(238),X(2)). See the note just above X(2106) in ETC for a list of points that lie on the 2nd equal-areas cubic. (The1st equal-areas cubic,Z(X(512),X(1)), 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 X

_{y}and X_{z}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, 1893by X

_{y}= y : z : x and X_{z}= z : x : y are a bicentric pair (if X is not the centroid) here named theof X, respectively. Casey calls the set {X, X1st and 2nd isobarycs_{y}, X_{z}} an isobaryc group of points and notes thatthe triangle formed by an isobaryc group is triply perspective to the reference triangle.If X = x : y : z (trilinears), then the isobarycs are given byX _{y}= by/a : cz/b : ax/c and X_{z}= 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, meansthe language of flowers. The symmedian point, X(6), also known as the Lemoine point in honor of Emile MichelHyacintheLemoine, has bicentrics P(6) and U(6), which may be called thehyacinth points.

## P(1) = 1st BROCARD POINT

f(a,b,c) = c/bThe 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

Paul Yiu,Introduction to the Geometry of the Triangle, 2002;

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 circleis the circle having center X(3) and radius eR, where e = (1 - 4 sin^{2}ω)^{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)P(1) and U(1) lie on the central line X(39)X(512), which is parallel to PU(2).

## P(2) = 1st BELTRAMI POINT

f(a,b,c) = a(b^{2}- a^{2})P(2) and U(2) are the circumcircle-inverses of P(1) and U(1), respectively.

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(3) = 1st YFF POINT

f(a,b,c) = bc[(c - u)/(b - u)]^{1/3}, where u is the (only) real root of the cubic polynomial

t^{3}+ (t - a)(t - b)(t - c)The bicentric pair P(3), U(3) are introduced and described in

Peter Yff,"An Analog of the Brocard Points,"American Mathematical Monthly70 (1963) 495-501;

see alsoMathWorld: Yff Points.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

f(a,b,c) = [cot B + cot C - 2 cot A + (tan B - tan C) L^{1/2}] sec A,

where L = - cot A cot B cot C (cot A + cot B + cot C)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(5) = 1st EHRMANN PIVOT

f(a,b,c) = 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(6) = 1st BICENTRIC OF X(6)

f(a,b,c) = bThese are the Brocardians (as mentioned at P(1)) of the incenter; they are also known as the

Jerabek pointsand theHyacinth points.P(6) and U(6) lie on the central line X(37)X(513).

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

## P(7) = 1st EVANS-YFF POINT

f(a,b,c) = bc(b^{4}- b^{2}c^{2}+ 2c^{2}a^{2}- 3a^{2}b^{2})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 trilin prod bary prod bicen sum bicen diff crosssum crossdiff tril pole ideal pt midpt 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 . . 523 5 567 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)

f(a,b,c) = 1/bP(8) and U(8) lie on the central line X(42)X(649).

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

f(a,b,c) = a/bP(9) and U(9) lie on the central line X(213)X(667).

## P(10) = TRILINEAR PRODUCT X(2)*P(8)

f(a,b,c) = b/aP(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(11) = ISOTOMIC CONJUGATE OF 1st BROCARD POINT

f(a,b,c) = b^{2}/aP(11) and U(11) lie on the central line X(141)X(523).

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

f(a,b,c) = a^{2}/bP(12) and U(12) lie on the central line X(1918)X(1919).

## P(13) = TRILINEAR SQUARE OF P(9)

f(a,b,c) = a^{2}/b^{2}P(13) and U(13) lie on the central line X(1923)X(1924).

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

f(a,b,c) = b^{2}/a^{2}P(14) and U(14) lie on the central line X(1577)X(1930).

Index trilin prod bary prod bicen sum bicen diff crosssum crossdiff tril pole ideal pt midpt 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 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 1932 1933 1934 1929 .

## P(15) = 1st BICENTRIC OF X(3)

f(a,b,c) = cos BP(15) and U(15) lie on the central line X(65)X(650).

## P(16) = 1st BICENTRIC OF X(4)

f(a,b,c) = sec BP(16) and U(16) lie on the central line X(73)X(652).

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

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

f(a,b,c) = cos B sec CP(17) and U(17) lie on the central line X(185)X(647).

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

f(a,b,c) = a cos BP(18) and U(18) lie on the central line X(663)X(1400).

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

f(a,b,c) = a sec B

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

f(a,b,c) = cos B csc AP(20) and U(20) lie on the central line X(226)X(522).

## P(21) = 1st BICENTRIC OF X(48)

f(a,b,c) = sin 2BP(21) and U(21) lie on the central line X(656)X(1953).

## P(22) = 1st BICENTRIC OF X(19)

f(a,b,c) = tan BP(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)

f(a,b,c) = cot BP(23) and U(23) lie on the central line X(31)X(661). The isogonal conjugates of PU(23) are PU(22).

Index trilin prod bary prod bicen sum bicen diff crosssum crossdiff tril pole ideal pt midpt 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 1946 . 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)

f(a,b,c) = 1 - b/aSuppose X = x : y : z is a triangle center other than the incenter. The

Vega transform of Xis defined in ETC (at X(1981)) as the point(y - z)/x : (z - x)/y : (x - y)/z. TheVega bicentrics of Xare 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(25) = ISOGONAL CONJUGATE OF P(24)

f(a,b,c) = a/(a - b)P(25) and U(25) lie on the circumcircle and on the line X(1015)X(1960).

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

f(a,b,c) = 1 - a/bP(26) and U(26) lie on the central line X(187)X(237), which is also P(2)U(2).

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

f(a,b,c) = b/(b - a)P(27) and U(27) lie on the central line X(244)X(665).

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

f(a,b,c) = (a + b - 2c)/(b - c)P(28) and U(28) lie on the central line X(1)X(6).

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

f(a,b,c) = 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)

f(a,b,c) = 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 trilin prod bary prod bicen sum bicen diff crosssum crossdiff tril pole ideal pt midpt 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 . . 1 1023 . 513 100 518 . 29 . . . 1983 . 523 110 . . 30 . . 1982 1981 . 647 648 30 .

## P(31) = 1st BICENTRIC OF X(37)

f(a,b,c) = a + cP(31) and U(31) lie on the central line X(513)X(1100), which is perpendicular to line X(1)X(3).

## P(32) = 1st BICENTRIC OF X(81)

f(a,b,c) = 1/(a + c)P(32) and U(32) lie on the central line X(661)X(1962).

## P(33) = 1st BICENTRIC OF X(513)

f(a,b,c) = a - cP(33) and U(33) lie on the central line X(44)X(513) (the antiorthic axis).

The line PU(33) is perpendicular to the line X(1)X(3).

## P(34) = 1st BICENTRIC OF X(100)

f(a,b,c) = 1/(a - c)P(34) and U(34) lie on the central line X(244)X(665), which is also P(27)U(27).

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(35) = 1st BICENTRIC OF X(31)

f(a,b,c) = b^{2}P(35) and U(35) lie on the central line X(38)X(661).

## P(36) = 1st BICENTRIC OF X(75)

f(a,b,c) = 1/b^{2}P(36) and U(36) lie on the central line X(513)X(1834).

Index trilin prod bary prod bicen sum bicen diff crosssum crossdiff tril pole ideal pt midpt 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 . 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)

f(a,b,c) = b/(a cos B)In

[Enumerative],objects 31-36 comprise ahexgenerated 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.P(37) and U(37) lie on the central line X(3)X(525).

The points PU(37) are isogonal conjugates of the points PU(39).

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

f(a,b,c) = (b cos B)/aSee 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(39) = 1st VERTEX PRODUCT OF HEX(110-115)

f(A,B,C) = a cot BSee 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).

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

f(a,b,c) = 1/[a(b^{2}- a^{2})]P(40) and U(40) lie on the central line X(115)X(125) and on the Steiner circumellipse.

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

f(a,b,c) = (a - b)c/aP(41) and U(41) lie on the line at infinity, X(30)X(511).

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(42) = ISOGONAL CONJUGATE OF P(41)

f(a,b,c) = a^{2}b/(a - b)P(42) and U(42) lie on the circumcircle.

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

Index trilin prod bary prod bicen sum bicen diff crosssum crossdiff tril pole ideal pt midpt 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

f(a,b,c) = bc[a^{2}(a^{2}+ 2b^{2}+ 3c^{2}) + b^{2}(c^{2}- b^{2})]P(43) is the centroid of the pedal triangle of the 1st Brocard point, P(1).

## P(44) = 1st LAEMMEL POINT

f(a,b,c) = sin B + (cos A sin C - sin A) cos B

R. Laemmelposed 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)

f(a,b,c) = c cos BP(45) and U(45) lie on the central line X(6)X(523).

## P(46) = 1st BICENTRIC OF X(9)

f(a,b,c) = a - b + cP(46) and U(46) lie on the central line X(6)X(513).

## P(47) = 1st BICENTRIC OF X(8)

f(a,b,c) = (a - b + c)/aP(47) and U(47) lie on the central line X(1)X(514).

## P(48) = 1st BICENTRIC OF X(55)

f(a,b,c) = (a - b + c)aP(48) and U(48) lie on the central line X(31)X(649).

## P(49) = 1st BICENTRIC OF X(514)

f(a,b,c) = (c - a)/bP(49) and U(49) lie on the central line X(663)X(672).

## P(50) = 1st BICENTRIC OF X(44)

f(a,b,c) = a - 2b + cP(50) and U(50) lie on the central line X(44)X(513).

Index trilin prod bary prod bicen sum bicen diff crosssum crossdiff tril pole ideal pt midpt 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 . . 1438 . . 49 101 692 672 663 . . . . . 50 88 106 44 513 . 1 1 513 44

## P(51) = 1st ACANTHUS POINT

f(a,b,c) = 2 - (b - c)/aP(51) and U(51) lie on the central line X(1)X(514).

## P(52) = 1st AMARANTH POINT

f(a,b,c) = 2b + cP(52) and U(52) lie on the central line X(37)X(513).

## P(53) = 1st ANGLICA POINT

f(a,b,c) = 2b - cP(53) and U(53) lie on the central line X(37)X(513).

## P(54) = 1st ANTHERICUM POINT

f(a,b,c) = 2a - b + cP(54) and U(54) lie on the central line X(6)X(513).

## P(55) = 1st ARUM POINT

f(a,b,c) = a + 2bP(55) and U(55) lie on the central line X(1)X(513).

## P(56) = 1st ASH POINT

f(a,b,c) = a - 2bP(56) and U(56) lie on the central line X(9)X(513).

## P(57) = 1st BICENTRIC OF X(650)

f(a,b,c) = cos A - cos CP(57) and U(57) lie on the central line X(44)X(513).

## P(58) = 1st BICENTRIC OF X(649)

f(a,b,c) = 1/c - 1/aP(58) and U(58) lie on the central line X(44)X(513).

## P(59) = 1st BICENTRIC OF X(56)

f(a,b,c) = 1 - cos B

## P(60) = 1st BICENTRIC OF X(40)

f(a,b,c) = cos C + cos A - cos B - 1P(60) and U(60) lie on the central line X(56)X(650).

## P(61) = 1st VACARETU POINT

f(a,b,c) = 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)

Index trilin prod bary prod bicen sum bicen diff crosssum crossdiff tril pole ideal pt midpt 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 . 650 . . . . . 60 84 1436 56 650 . . . . . 61 . . 1594 523 . . . 523 1594

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

f(a,b,c) = a(a^{2}- b^{2})(a^{2}+ c^{2}- 2b^{2})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)

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

f(a,b,c) = (a^{2}- b^{2}- c^{2})S(a,b,c) - 4(b^{2}- c^{2})( b^{2}+ c^{2}- 2a^{2})σ,

where S(a,b,c) = a^{4}+ b^{4}+ c^{4}- b^{2}c^{2}- c^{2}a^{2}- a^{2}b^{2}, 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)

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

f(a,b,c) = a^{2}[bcS(a,b,c) - 2(b^{2}- c^{2})(2a^{2}- b^{2}- c^{2})σ],

where S(a,b,c) = a^{4}+ b^{4}+ c^{4}- b^{2}c^{2}- c^{2}a^{2}- a^{2}b^{2}, 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)

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

f(a,b,c) = [a^{2}(b^{2}+ c^{2}) - (b^{2}+ c^{2})^{2}]S(a,b,c) - 4a^{2}(b^{2}- c^{2})(2a^{2}- c^{2})σ,

where S(a,b,c) = a^{4}+ b^{4}+ c^{4}- b^{2}c^{2}- c^{2}a^{2}- a^{2}b^{2}, 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)

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

f(a,b,c) = a(b + c)[bcS(a,b,c) - 2a(b - c)(2a^{2}- b^{2}- c^{2})σ],

where S(a,b,c) = a^{4}+ b^{4}+ c^{4}- b^{2}c^{2}- c^{2}a^{2}- a^{2}b^{2}, 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)

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

f(a,b,c) = a^{2}(b^{2}+ c^{2})S(a,b,c) - 4a^{2}(b^{2}- c^{2})(2a^{2}- b^{2}- c^{2})σ,

where S(a,b,c) = a^{4}+ b^{4}+ c^{4}- b^{2}c^{2}- c^{2}a^{2}- a^{2}b^{2}, 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)

Index trilin prod bary prod bicen sum bicen diff crosssum crossdiff tril pole ideal pt midpt 62 . . . 351 . 543 . . . 63 . . 63 2642 . . . . . 64 . . . . . . . . . 65 . . 6 . . . . . . 66 . . 37 . . . . . . 67 . . . . . . . . .

## P(68) = 1st BICENTRIC OF X(5)

f(a,b,c) = cos(C - A)P(68) and U(68) lie on the central line X(654)X(2594).

## P(69) = 1st BICENTRIC OF X(54)

f(a,b,c) = sec(C - A)P(69) and U(69) lie on the central line X(2599)X(2600).

## P(70) = 1st BICENTRIC OF X(110)

f(a,b,c) = sin(C - A)P(70) and U(70) lie on the central line X(2245)X(2605).

## P(71) = 1st BICENTRIC OF X(523)

f(a,b,c) = csc(C - A)P(71) and U(71) lie on the central line X(2610)X(2611).

## P(72) = 1st BICENTRIC OF X(2620)

f(a,b,c) = tan(C - A)P(72) and U(72) lie on the central line X(2290)X(2618).

## P(73) = 1st BICENTRIC OF X(2621)

f(a,b,c) = cot(C - A)P(73) and U(73) lie on the central line X(1109)X(2624).

## P(74) = 1st BICENTRIC OF X(656)

f(a,b,c) = tan C - tan AP(74) and U(74) lie on the central line X(44)X(513).

## P(75) = 1st BICENTRIC OF X(162)

f(a,b,c) = 1/(tan C - tan A)P(75) and U(75) lie on the central line X(2631)X(2632).

Index trilin prod bary prod bicen sum bicen diff crosssum crossdiff tril pole ideal pt midpt 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 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)

f(a,b,c) = sec C - sec AP(76) and U(76) lie on the central line X(44)X(513).

## P(77) = 1st BICENTRIC OF X(653)

f(a,b,c) = 1/(sec C - sec A)P(77) and U(77) lie on the central line X(2637)X(2638).

## P(78) = 1st BICENTRIC OF X(661)

f(a,b,c) = cot C - cot AP(78) and U(78) lie on the central line X(44)X(513).

## P(79) = 1st BICENTRIC OF X(662)

f(a,b,c) = 1/(cot C - cot A)P(79) and U(79) lie on the central line X(2643)X(2643).

## P(80) = 1st BICENTRIC OF X(65)

f(a,b,c) = cos C + cos AP(80) and U(80) lie on the central line X(650)X(2646).

## P(81) = 1st BICENTRIC OF X(21)

f(a,b,c) = 1/(cos C + cos A)P(81) and U(81) lie on the central line X(661)X(2650).

## P(82) = 1st BICENTRIC OF X(73)

f(a,b,c) = sec C + sec AP(82) and U(82) lie on the central line X(652)X(2654).

## P(83) = 1st BICENTRIC OF X(29)

f(a,b,c) = 1/(sec C + sec A)P(83) and U(83) lie on the central line X(822)X(2658).

## P(84) = 1st BICENTRIC OF X(42)

f(a,b,c) = 1/c + 1/aP(84) and U(84) lie on the central line X(649)X(2666).

## P(85) = 1st BICENTRIC OF X(86)

f(a,b,c) = ac/(a + c)P(85) and U(85) lie on the central line X(798)X(2667).

Index trilin prod bary prod bicen sum bicen diff crosssum crossdiff tril pole ideal pt midpt 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 . 649 2663 2664 2665 . 2666 85 42 213 2667 798 2668 2669 2107 . 2670

## P(86) = 1st BICENTRIC OF X(30)

f(a,b,c) = ca[2b^{4}- (c^{2}-a^{2})^{2}- b^{2}(c^{2}+ a^{2})]P(86) and U(86) lie on the line at infinity.

## P(87) = TRILINEAR PRODUCT P(9)*P(86)

f(a,b,c) = a/[2b^{4}- (c^{2}-a^{2})^{2}- b^{2}(c^{2}+ a^{2})]P(87) and U(87) lie on the circumcircle.

## P(88) = 1st BICENTRIC OF X(511)

f(a,b,c) = b(b^{2}c^{2}+ b^{2}a^{2}- c^{4}- a^{4})P(88) and U(88) lie on the line at infinity.

## P(89) = TRILINEAR PRODUCT P(9)*P(88)

f(a,b,c) = a/b^{2}c^{2}(b^{2}c^{2}+ b^{2}a^{2}- c^{4}- a^{4})P(89) and U(89) lie on the circumcircle.

## P(90) = 1st BICENTRIC OF X(512)

f(a,b,c) = b(c^{2}- a^{2})P(90) and U(90) lie on the line at infinity.

## P(91) = TRILINEAR PRODUCT P(9)*P(90)

a/(b^{2}(c^{2}- a^{2}))P(91) and U(91) lie on the circumcircle.

## P(92) = 1st BICENTRIC OF X(513)

f(a,b,c) = c-aP(92) and U(92) lie on the line at infinity.

## P(93) = TRILINEAR PRODUCT P(9)*P(92)

a/(bc - ba))P(93) and U(93) lie on the circumcircle.

## P(94) = 1st BICENTRIC OF X(514)

ca(c-a)P(94) and U(94) lie on the line at infinity.

## P(95) = TRILINEAR PRODUCT P(9)*P(94)

a/(c - a)P(95) and U(95) lie on the circumcircle.

## P(96) = 1st BICENTRIC OF X(518)

bc + ba - c^{2}- a^{2}P(96) and U(96) lie on the line at infinity.

## P(97) = TRILINEAR PRODUCT P(9)*P(96)

a/(b(bc + ba - c^{2}- a^{2}))P(97) and U(97) lie on the circumcircle.

## P(98) = 1st BICENTRIC OF X(519)

f(a,b,c) = ca(2b - c - a)P(98) and U(98) lie on the line at infinity.

## P(99) = TRILINEAR PRODUCT P(9)*P(98)

f(a,b,c) = a/(2 - c - a)P(99) and U(99) lie on the circumcircle.

## P(100) = 1st BICENTRIC OF X(521)

f(A,B,C) = (sec C - sec A)/bP(100) and U(100) lie on the line at infinity.

## P(101) = TRILINEAR PRODUCT P(9)*P(100)

f(a,b,c) = a/(sec C - sec A)P(101) and U(101) lie on the circumcircle.

## P(102) = 1st BICENTRIC OF X(522)

f(a,b,c) = (cos C - cos A)/bP(102) and U(102) lie on the line at infinity.

## P(103) = TRILINEAR PRODUCT P(9)*P(102)

f(a,b,c) = a/(cos C - cos A)P(103) and U(103) lie on the circumcircle.

## P(104) = 1st BICENTRIC OF X(523)

f(a,b,c) = (c^{2}- a^{2})/bP(104) and U(104) lie on the line at infinity.

## P(105) = TRILINEAR PRODUCT P(9)*P(104)

f(a,b,c) = a/(c^{2}- a^{2})P(105) and U(105) lie on the circumcircle.

## P(106) = 1st BICENTRIC OF X(524)

f(a,b,c) = (2b^{2}- c^{2}- a^{2})/bP(106) and U(106) lie on the line at infinity.

## P(107) = TRILINEAR PRODUCT P(9)*P(106)

f(a,b,c) = a/(2b^{2}- c^{2}- a^{2})P(107) and U(107) lie on the circumcircle.

## P(108) = 1st BICENTRIC OF X(525)

f(a,b,c) = (c cos C - a cos A)/bP(108) and U(108) lie on the line at infinity.

## P(109) = TRILINEAR PRODUCT P(9)*P(108)

f(a,b,c) = a/(c cos C - a cos A)P(109) and U(109) lie on the circumcircle.

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

f(a,b,c) = a^{2}(b - 2c) - a(b^{2}+ bc - c^{2}) - (b - c)c^{2}Let A'B'C' be the cevian triangle of X(1). Let A

_{B}be the reflection of A' in BB', and define B_{C}and C_{A}cyclically. Let A_{C}be the reflection of A' in BC', and define B_{A}and C_{B}cyclically. Let O_{AB}be the circumcircle of AA'A_{B}, and define O_{BC}and O_{CA}cyclically. Let O_{AC}be the circumcircle of AA'A_{C}, and define O_{BA}and O_{CB}cyclically. Then P(110) is the radical center of O_{AB}, O_{BC}, O_{CA}, and U(110) is defined symmetrically; i.e., as the radical center of O_{AC}, O_{BA}, O_{CB}. Angel Montesdeoca, August 26 2013See Hechos Geometricos 240813 and Anopolis 885

P(110) and U(110) lie on the line X(942)X(1938); P(110)U(110) has ideal point X(513).

## P(111) = 2nd MONTESDEOCA NINE-POINT CIRCLES RADICAL CENTER

f(a,b,c) = (a + b - c)(a - b + c)(a^{3}b + 2a^{3}c + a^{2}c^{2}- ab^{3}- 2ac^{3}- 2ab^{2}c - 3abc^{2}- 3bc^{3}- 3b^{2}c^{2}- b^{3}c - c^{4})Let A'B'C' be the cevian triangle of X(1). Let A

_{B}be the reflection of A' in BB', and define B_{C}and C_{A}cyclically. Let A_{C}be the reflection of A' in BC', and define B_{A}and C_{B}cyclically. Let N_{AB}be the nine-point circle of AA'A_{B}, and define N_{BC}and N_{CA}cyclically. Let N_{AC}be the nine-point circle of AA'A_{C}, and define N_{BA}and N_{CB}cyclically. Then P(111) is the radical center of N_{AB}, N_{BC}, N_{CA}, and U(111) is defined symmetrically; i.e., as the radical center of N_{AC}, N_{BA}, N_{CB}. Angel Montesdeoca, August 26 2013See Hechos Geometricos 240813 and Anopolis 860

First edition, PU(1) to PU(42): 9/22/03. Second edition, to PU(85): 8/22/04. Third edition, to PU(111): 9/01/13.

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