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 the homogeneity requirement (1) in the definition of triangle center (in Glossary), but, unlike the symmetry requirement (2), suppose that that |f(a,b,c)| is not equal to |f(a,c,b)|. Thenf(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 byc/b : a/c : b/a and b/c : c/a : a/b. A geometric interpretation of the (strictly algebraic) requirement that |f(a,b,c)| be not equal to |f(a,c,b)| for a point P = f(a,b,c) : f(b,c,a) : f(c,a,b) is that in any list of steps for constructing P, interchanging B and C results in a point other than P. An equivalent and more precise version by César Eliud Lozada (April 14, 2024) follows:Let ABC be a triangle, with respective opposite sidelengths a, b, c, and in the plane of ABC, let P be a point having homogeneous barycentric coordinates f(a, b, c) : f(b, c, a) : f(c, a, b). Let Am be the midpoint of segment BC, let A' be the reflection of A in Am, so that A'BC is inversely similar to ABC. Let P' = P-of-A'BC, taking into account that in A'BC the sidelengths opposite A', B, C, are a, c, b, respectively. (To construct a point X from its signed distances, x,y,z from the respective sides BC, CA, AB, let Lb be the line parallel to AC at signed distance y from AC, and let Lc be the line parallel to AB at signed distance z from AB. Then P = Lb∩Lc.) If the point P' is the reflection of P in Am, then P is a triangle center. Otherwise, let U be the reflection of P' in Am. Then P and U are a bicentric pair of points of ABC. (Throughout, it is understood that ABC is an arbitrary triangle, which is to say that the descriptions must hold for all triangles.)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 coordinate f(a,c,b). (The coordinates may be either trilinnear or barycentric.) The two tables just below give definitions of operations that carry a bicentric pair P = p : q : r and U = u : v : w to a triangle center. For two of these operations (bicentric difference and bicentric sum), the domain is restricted to points P for which there exists a polynomial p(a,b,c) such that the first barycentric coordinate of P is p(a,b,c). Specifically, to avoid ambiguity (as in Example 2 below), we assume that p,q,r and u,v,w are represented byp(a,b,c), p(b,c,a), p(c,a,b) and p(a,c,b), p(b,a,c), p(c,b,a), where p(a,b,c) is a polynomial in (a,b,c), and GCD(p(a,b,c), p(b,c,a)) = 1. Example 1. P = p : q : r, where p = f(a,b,c) = a^2 (a^2 - b^2)(b^2 - c^2). Here, (barycentric difference of PU) = p-u : q-v : r-w = a^2 (b^2 - c^2)(c^2 - b^2) : : = X(351). Example 2. P = p : q : r, where p = g(a,b,c) = a^2/(c^2 - a^2), so that P is the same point as in Example 1, but given by a nonpolynomial representation, leading to p-u : q-v : r-w = a^2(b^2 - c^2)(2a^2 - b^2 - c^2) : : , which is X(3124), not X(351). To conclude, the representation in Example 1 is required, whereas the nonpolynomial representation for the same point, in Example 2, must be replaced when formulating the bicentric difference P-U and bicentric sum P+U.
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/b^2 : 1/c^2 : 1/a^2The 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 circle is the circle having center X(3) and radius eR, where e = (1 - 4 sin^2ω)^(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*(a^2-b^2) : b*(b^2-c^2) : c*(c^2-a^2) : :
Barycentrics (a^2-b^2)*a^2 : (b^2-c^2)*b^2 : (c^2-a^2)*c^2 : :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 b*c*((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)
Barycentrics ((c - u)/(b - u))^(1/3) : :
Barycentrics
P(3) = (f(a, b, c)/f(a, c, b))^(1/3) : :
U(3) = (f(a, c, b)/f(a, b, c))^(1/3) : :
where:
f(a, b, c) = L^2+(a+b-5*c)*L+b^2-4*b*c+c^2+a^2-4*a*(b+c)
L = (a^3-6*(b+c)*a^2-3*(2*b^2-11*b*c+2*c^2)*a-6*b*c^2+c^3+b^3-6*b^2*c+3*K)^(1/3)
K = (-3*(b-c)^2*a^4+6*(b+c)*(3*b^2-7*b*c+3*c^2)*a^3-3*(b^4+c^4+b*c*(8*b^2-57*b*c+8*c^2))*a^2+6*(b+c)*(b^2-5*b*c+c^2)*b*c*a-3*b^2*c^2*(b^2-6*b*c+c^2))^(1/2)
(César Lozada - Apr 8, 2024)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;
see also MathWorld: 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
Trilinears (cot(B) + cot(C) - 2*cot(A) + (tan(B) - tan(C))*sqrt(L))*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))*sqrt(L))*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) : :
Barycentrics -2*(b^2-c^2)*S+sqrt(3)*((b^2+c^2)*a^2-(b^2-c^2)^2) : : (César Lozada - Apr 11, 2024)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 ((3*b^2-2*c^2)*a^2-b^4+b^2*c^2)/a : :
Barycentrics (3*b^2-2*c^2)*a^2-b^4+b^2*c^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).
P(8) = 1st BICENTRIC OF X(2)
Trilinears 1/b : 1/c : 1/a
Barycentrics a/b : b/c : c/aP(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 a^3*c : b^3*a : c^3*bP(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 : 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(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 b^2/a : c^2/b : a^2/c
Barycentrics b^2 : c^2 : a^2P(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 a^2/b : b^2/c : c^2/a
Barycentrics a^3/b : b^3/c : c^3/aP(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 a^2/b^2 : b^2/c^2 : c^2/a^2
Barycentrics a^3/b^2 : b^3/c^2 : c^3/a^2P(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 b^2/a^2 : c^2/b^2 : a^2/c^2
Barycentrics b^2/a : c^2/b : a^2/cP(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)
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)
Barycentrics (a^2-b^2+c^2)/c : : (César Lozada - Apr 11, 2024)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)
Barycentrics a*(a^2+b^2-c^2)*(-a^2+b^2+c^2)/b : : (César Lozada - Apr 8, 2024)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) : sin(C)*cos(A)*sec(B)
Barycentrics (a^2-b^2+c^2)^2*(-a^2+b^2+c^2)/c^2 : : (César Lozada - Apr 8, 2024)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)*sin(A) : cos(C)*sin(B) : cos(A)*sin(C)
Barycentrics cos(B)*sin(A)^2 : cos(C)*sin(B)^2 : cos(A)*sin(C)^2
Barycentrics a*(a^2-b^2+c^2)/c : : (César Lozada - Apr 8, 2024)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 a^2*sec(B) : b^2*sec(C) : c^2*sec(A)
Barycentrics a^2*(a^2+b^2-c^2)*(-a^2+b^2+c^2)/b : : (César Lozada - Apr 8, 2024)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)
Barycentrics b*(a^2-b^2+c^2) : : (César Lozada - Apr 8, 2024)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(2*B) : sin(2*C) : sin(2*A)
Barycentrics sin(A)*sin(2*B) : sin(B)*sin(2*C) : sin(C)*sin(2*A)
Barycentrics a*b^2*(a^2-b^2+c^2) : : (César Lozada - Apr 8, 2024)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)
Barycentrics a*(a^2+b^2-c^2)*(-a^2+b^2+c^2) : : (César Lozada - Apr 8, 2024)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)
Barycentrics a*(a^2-b^2+c^2) : : (César Lozada - Apr 8, 2024)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).
P(24) = 1st VEGA BICENTRIC OF X(6)
Trilinears (a - b)/a : (b - c)/b : (c - a)/c
Barycentrics a - b : b - c : c - aSuppose 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 a^2/(a - b) : b^2/(b - c) : c^2/(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 (a-b)/b : (b-c)/c : (c-a)/a
Barycentrics a*(a-b)/b : b*(b-c)/c : c*(c-a)/aP(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 a*b/(b - a) : b*c/(c - b) : c*a/(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)*(a-c)*(a+b-2*c) : (b-c)*(b-a)*(b+c-2*a) : (c-a)*(c-b)*(a-2*b+c)
Barycentrics a*(a-b)*(a-c)*(a+b-2*c) : b*(b-c)*(b-a)*(b+c-2*a) : c*(c-a)*(c-b)*(a-2*b+c)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)) : :
Barycentrics c*a^2*(a+b)*(a^2-b^2)*(a^2-c^2)*(a^2-b*a+b^2-c^2) : : (César Lozada - Apr 8, 2024)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))) : :
Barycentrics (a+b)*(a^2-b^2)*(a^2-c^2)*(a^2+b^2-c^2)*(a^2-b^2+c^2)*(b*a^3-(2*b^2-c^2)*a^2+(b^2-c^2)*b*a+(b^2-c^2)*c^2) : : (César Lozada - Apr 8, 2024)P(30) and U(30) lie on the Euler line, X(2)X(3).
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 b^2 : c^2 : a^2
Barycentrics a*b^2 : b*c^2 : c*a^2P(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/b^2 : 1/c^2 : 1/a^2
Barycentrics a/b^2 : b/c^2 : c/a^2P(36) and U(36) lie on the central line X(513)X(1834).
P(36) = isogonal conjugate of P(35)
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)
Barycentrics (a^2+b^2-c^2)*(-a^2+b^2+c^2) : : (César Lozada - Apr 8, 2024)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(2*B) : sin(2*C) : sin(2*A)
Barycentrics b^2*(a^2-b^2+c^2) : : (César Lozada - Apr 8, 2024)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 a^2*cot(B) : b^2*cot(C) : c^2*cot(A)
Barycentrics a^2*(a^2-b^2+c^2) : : (César Lozada - Apr 8, 2024)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 (b^2-c^2)*(a^2-c^2)/a : (c^2-a^2)*(b^2-a^2)/b : (a^2-b^2)*(c^2-b^2)/c
Barycentrics (b^2-c^2)*(a^2-c^2) : (c^2-a^2)*(b^2-a^2) : (a^2-b^2)*(c^2-b^2)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)*a/b : (c-a)*b/c
Barycentrics (a-b)*c : (b-c)*a : (c-a)*bP(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 a*(b-c)*(a-c)/c : b*(c-a)*(b-a)/a : c*(a-b)*(c-b)/b
Barycentrics a^2*(b-c)*(a-c)/c : b^2*(c-a)*(b-a)/a : c^2*(a-b)*(c-b)/bP(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)
P(43) = 1st ACACIA POINT
Trilinears (a^4+(2*b^2+3*c^2)*a^2-(b^2-c^2)*b^2)/a : :
Barycentrics a^4+(2*b^2+3*c^2)*a^2-(b^2-c^2)*b^2 : :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)) : :
Barycentrics a*(a^3+(b-c)*a^2-(3*b^2-c^2)*a+(b-c)*(b^2+c^2)) : : (César Lozada - Apr 8, 2024)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 (a - b + c)/a : (b - c + a)/b : (c - a + b)/c
Barycentrics a - b + c : b - c + a : c - a + bP(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^2P(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)/aP(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 - 2*b + c : b - 2*c + a : c - 2*a + b
Barycentrics a*(a - 2*b + c) : b*(b - 2*c + a) : c*(c - 2*a + 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)
P(51) = 1st ACANTHUS POINT
Trilinears (2*a - b + c)/a : :
Barycentrics 2*a - b + c : :P(51) and U(51) lie on the central line X(1)X(514).
P(52) = 1st AMARANTH POINT
Trilinears 2*b + c : 2*c + a : 2*a + b
Barycentrics a*(2*b + c) : b*(2*c + a) : c*(2*a + 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 2*b - c : 2*c - a : 2*a - b
Barycentrics a*(2*b - c) : b*(2*c - a) : c*(2*a - 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 2*a - b + c : 2*b - c + a : 2*c - a + b
Barycentrics a*(2*a - b + c) : b*(2*b - c + a) : c*(2*c - 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 + 2*b : b + 2*c : c + 2*a
Barycentrics a*(a + 2*b) : b*(b + 2*c) : c*(c + 2*a)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 - 2*b : b - 2*c : c - 2*a
Barycentrics a*(a - 2*b) : b*(b - 2*c) : c*(c - 2*a)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/bP(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))
Barycentrics (a+b-c)*(-a+b+c)/c : : (César Lozada - Apr 8, 2024)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) : :
Barycentrics a*(a^3+a^2*(b-c)-a*(b-c)^2-(b+c)*(b^2-c^2)) : : (César Lozada - Apr 8, 2024)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) : :
Barycentrics (a^4*b^2+(b^2-c^2)^3-a^2*(b^2-c^2)*(2*b^2+c^2))*(a^4-c^2*(b^2-c^2)-a^2*(b^2+2*c^2)) : : (César Lozada - Apr 8, 2024)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)
P(62) = INSIMILICENTER OF 2nd LEMOINE AND PARRY CIRCLES
Trilinears a*(a^2-b^2)*(a^2-2*b^2+c^2) : :
Barycentrics a^2*(a^2-b^2)*(a^2-2*b^2+c^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)
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*(2*(2*a^2-b^2-c^2)*(b^2-c^2)*S-(-a^2+b^2+c^2)*(a^4-(b^2+c^2)*a^2+b^4-b^2*c^2+c^4)) : : ,
Barycentrics a^2*((-a^2+b^2+c^2)*(a^4+b^4-b^2*c^2+c^4-a^2*(b^2+c^2))-2*(2*a^2-b^2-c^2)*(b^2-c^2)*S) : :, where S=2*Area(ABC) (César Lozada - Apr 8, 2024)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*((2*a^2-b^2-c^2)*(b^2-c^2)*S-b*c*(a^4-(b^2+c^2)*a^2+b^4-b^2*c^2+c^4)) : :
Barycentrics a^2*((2*a^2-b^2-c^2)*(b^2-c^2)*S-b*c*(a^4-(b^2+c^2)*a^2+b^4-b^2*c^2+c^4)) : :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 (2*S*a^2*(b^2-c^2)*(2*a^2-b^2-c^2)-((b^2+c^2)*a^2-(b^2-c^2)^2)*(a^4-(b^2+c^2)*a^2+b^4-b^2*c^2+c^4))/a : :
Barycentrics 2*S*a^2*(b^2-c^2)*(2*a^2-b^2-c^2)-((b^2+c^2)*a^2-(b^2-c^2)^2)*(a^4-(b^2+c^2)*a^2+b^4-b^2*c^2+c^4) : : (César Lozada - Apr 8, 2024)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 (b+c)*(b*c*(a^4+b^4-b^2*c^2+c^4-a^2*(b^2+c^2))-a*(b-c)*(2*a^2-b^2-c^2)*S) : :,
Barycentrics a*(b+c)*(b*c*(a^4+b^4-b^2*c^2+c^4-a^2*(b^2+c^2))-a*(b-c)*(2*a^2-b^2-c^2)*S) : : (César Lozada - Apr 8, 2024)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 a*(2*(2*a^2-b^2-c^2)*(b^2-c^2)*S-(b^2+c^2)*(a^4-(b^2+c^2)*a^2+b^4-b^2*c^2+c^4)) : :
Barycentrics a^2*(2*(2*a^2-b^2-c^2)*(b^2-c^2)*S-(b^2+c^2)*(a^4-(b^2+c^2)*a^2+b^4-b^2*c^2+c^4)) : : (César Lozada - Apr 8, 2024)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)
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)
Barycentrics a*(a^4-c^2*(b^2-c^2)-a^2*(b^2+2*c^2))/b : : (César Lozada - Apr 8, 2024)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)
Barycentrics (a^4-(2*b^2+c^2)*a^2+(b^2-c^2)*b^2)*((b^2+c^2)*a^2-(b^2-c^2)^2)/c : : (César Lozada - Apr 8, 2024)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 (a^2 - c^2)/b : :
Barycentrics sin(A)*sin(C - A) : sin(B)*sin(A - B) : sin(C)*sin(B - C)
Barycentrics a*(a^2-c^2)/b : : (César Lozada - Apr 8, 2024)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).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)
Barycentrics (a^2-b^2)*(b^2-c^2)/c : : (César Lozada - Apr 8, 2024)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)
Barycentrics a*(a^4-(2*b^2+c^2)*a^2+(b^2-c^2)*b^2)*((b^2+c^2)*a^2-(b^2-c^2)^2)*(a^2-c^2) : : (César Lozada - Apr 8, 2024)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)
Barycentrics a*(a^2-b^2)*(b^2-c^2)*(a^4-(b^2+2*c^2)*a^2-(b^2-c^2)*c^2) : : (César Lozada - Apr 8, 2024)P(73) and U(73) lie on the central line X(1109)X(2624).
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))
Barycentrics a*(a^2-c^2)*(a^2-b^2+c^2) : : (César Lozada - Apr 8, 2024)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(A) - tan(C)) : 1/(tan(B) - tan(A)) : 1/(tan(C) - tan(B))
Barycentrics sin(A)/(tan(A) - tan(C)) : sin(B)/(tan(B) - tan(A)) : sin(C)/(tan(C) - tan(B))
Barycentrics a*(a^2-b^2)*(b^2-c^2)*(a^2+b^2-c^2)*(-a^2+b^2+c^2) : : (César Lozada - Apr 8, 2024)P(75) and U(75) lie on the central line X(2631)X(2632).
P(75) = isogonal conjugate of P(74)
P(76) = 1st BICENTRIC OF X(652)
Trilinears sec(A) - sec(C) : :
Barycentrics sin(A)*(sec(A) - sec(C)) : :
Barycentrics (a-c)*(a-b+c)*(a^2-b^2+c^2)/c : : (César Lozada - Apr 8, 2024)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(A) - sec(C)) : :
Barycentrics sin(A)/(sec(A) - sec(C)) : :
Barycentrics a*(a-b)*(b-c)*(a+b-c)*(-a+b+c)*(a^2+b^2-c^2)*(-a^2+b^2+c^2)/b : : (César Lozada - Apr 8, 2024)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(A) - cot(C) : :
Trilinears a^2 - c^2 : :
Barycentrics sin(A)*(cot(A) - cot(C)) : :
Barycentrics a*(a^2-c^2) : : (César Lozada - Apr 8, 2024)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(A) - cot(C)) : :
Barycentrics sin(A)/(cot(A) - cot(C)) : :
Barycentrics a*(a^2-b^2)*(b^2-c^2) : : (César Lozada - Apr 8, 2024)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(A) + cos(C) : :
Barycentrics sin(A)*cos(A) + cos(C)) : :
Barycentrics a*(a+c)*(a+b-c)*(-a+b+c) : : (César Lozada - Apr 8, 2024)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(A) + cos(C)) : :
Barycentrics sin(A)/( cos(A) + cos(C)) : :
Barycentrics a*(a+b)*(b+c)*(a-b+c) : : (César Lozada - Apr 8, 2024)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(A) + sec(C) : :
Barycentrics sin(A)*sec(A) + sec(C)) : :
Barycentrics (a+c)*(a+b-c)*(-a+b+c)*(a^2-b^2+c^2)/c : : (César Lozada - Apr 8, 2024)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(A) + sec(C)) : :
Barycentrics sin(A)/(sec(A) + sec(C)) : :
Barycentrics a*(a+b)*(b+c)*(a-b+c)*(a^2+b^2-c^2)*(-a^2+b^2+c^2)/b : : (César Lozada - Apr 8, 2024)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 b*(a+c) : :
Barycentrics (a+c)/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 a*c/(a + c) : :
Barycentrics a^2*c/(a + c) : :P(85) and U(85) lie on the central line X(798)X(2667).
P(85) = isogonal conjugate of P(84)
P(86) = 1st BICENTRIC OF X(30)
Trilinears (2*b^4-(a^2+c^2)*b^2-(a^2-c^2)^2)/b : :The points P(86) and U(86) lie on the central 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^4-(2*b^2-c^2)*a^2+(b^2-c^2)*(b^2+2*c^2))*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*a : :P(87) and U(87) lie on the circumcircle.
P(88) = 1st BICENTRIC OF X(511)
Trilinears b*(a^4-b^2*a^2-(b^2-c^2)*c^2) : :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^4-c^2*a^2+(b^2-c^2)*b^2)*((b^2+c^2)*a^2-b^4-c^4)*a/b^2 : :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*(c^2 - a^2) : :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^2-b^2)*(b^2-c^2)*a/b^2 : :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*(a - b + c)/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 a^2/(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 b*(a - b + c) : :
Barycentrics (a - b + c)/c : :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*(-a+b+c)*(a+b-c)/b^2 : :
Barycentrics a^2*(-a+b+c)*(a+b-c)/b^2 : : : :P(95), U(95), and X(8638) are collinear.
P(96) = 1st BICENTRIC OF X(518)
Trilinears b*c + b*a - c^2 - a^2 : :
Barycentrics a*(b*c + b*a - c^2 - a^2) : :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*(b*c + b*a - c^2 - a^2)) : :
Barycentrics a^2/(b*(b*c + b*a - c^2 - a^2)) : :P(97) and U(97) lie on the circumcircle.
P(98) = 1st BICENTRIC OF X(519)
Trilinears a*c*(2*b - c - a) : :
Barycentrics a^2*c*(2*b - 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 a/(2*b-a-c) : :
Barycentrics a^2/(2*b-a-c) : :P(99) and U(99) lie on the circumcircle and the central line X(1017)X(1960).
P(100) = 1st BICENTRIC OF X(521)
Trilinears (sec(A) - sec(C))/b : :
Barycentrics a*(sec(A) - sec(C))/b : :
Barycentrics a*(a-c)*(a-b+c)*(a^2-b^2+c^2) : : (César Lozada - Apr 8, 2024)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).
P(101) = TRILINEAR QUOTIENT P(9)/P(100)
Trilinears a/(sec(A) - sec(C)) : :
Barycentrics a^2/(sec(A) - sec(C)) : :
Barycentrics a^2*(a-b)*(b-c)*(a+b-c)*(-a+b+c)*(a^2+b^2-c^2)*(-a^2+b^2+c^2)/b : : (César Lozada - Apr 8, 2024)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(A) - cos(C) : :
Barycentrics sin(A)*(cos(A) - cos(C)) : :
Barycentrics a*(a-c)*(a-b+c)/b : : (César Lozada - Apr 8, 2024)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).
P(103) = TRILINEAR QUOTIENT P(9)/P(102)
Trilinears a/(cos(A) - cos(C)) : :
Barycentrics a^2/(cos(A) - cos(C)) : :
Barycentrics a^2*(a-b)*(b-c)*(a+b-c)*(-a+b+c) : : (César Lozada - Apr 8, 2024)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 a*c/(c + a - b) : :
Barycentrics a^2*c/(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/(c^2 - a^2) : :
Barycentrics a^2/(c^2 - a^2) : :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 (2*b^2 - c^2 - a^2)/b : :
Barycentrics a*(2*b^2 - c^2 - a^2)/b : :P(106) lies on non-central line X(238)P(26).
P(107) = TRILINEAR QUOTIENT P(9)/P(106)
Trilinears a/(2*b^2 - c^2 - a^2) : :
Barycentrics a^2/(2*b^2 - c^2 - a^2) : :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 (a*cos(A) - c*cos(C))/b : :
Barycentrics a*(a*cos(A) - c*cos(C))/b : :
Barycentrics a*(a^2-c^2)*(a^2-b^2+c^2)/b : : (César Lozada - Apr 8, 2024)P(108) lies on non-central line X(238)P(26).
P(109) = TRILINEAR QUOTIENT P(9)/P(108)
Trilinears a/(a*cos(A) - c*cos(C)) : :
Barycentrics a^2/(a*cos(A) - c*cos(C)) : :
Barycentrics a^2*(a^2-b^2)*(b^2-c^2)*(a^2+b^2-c^2)*(-a^2+b^2+c^2) : : (César Lozada - Apr 8, 2024)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 a^2*(b - 2*c) - a*(b^2 + b*c - c^2) - (b - c)*c^2 : :
Barycentrics a*(a^2*(b - 2*c) - a*(b^2 + b*c - c^2) - (b - c)*c^2) : :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)
See 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(1938).
P(111) = 1st MONTESDEOCA NINE-POINT CIRCLES RADICAL CENTER
Trilinears (a+b-c)*(a-b+c)*((b+2*c)*a^3+c^2*a^2-(b+c)*(b^2+b*c+2*c^2)*a-c*(b+c)^3) : :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)
See Hechos Geometricos 240813 and Anopolis 860
The ideal point of P111) is X(513).
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*(b^2 - c^2)*(a^4 - b^2*c^2 - a^2*Z) + (V - W)^(1/2) : : , where
Z = (a^4 + b^4 + c^4 - b^2*c^2 - c^2*a^2 - a^2*b^2)^(1/2)
V = 2a^2*b^2*c^2*Z^3
W = b^6*c^6 + c^6*a^6 + a^6*b^6 - 3*a^4*b^4*c^4 - (b^4*c^4 + c^4*a^4 + a^4*b^4)*Z^2PU(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(b^2 - c^2)(a^4 - b^2c^2 - a^2Z) - (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*b^2 - c^2)*(a^4 - b^2*c^2 + a^2*Z) + (- V - W)^(1/2) : : , where
Z = (a^4 + b^4 + c^4 - b^2*c^2 - c^2*a^2 - a^2*b^2)^(1/2)
V = 2*a^2*b^2*c^2*Z^3
W = b^6*c^6 + c^6*a^6 + a^6*b^6 - 3*a^4*b^4*c^4 - (b^4*c^4 + c^4*a^4 + a^4*b^4)*Z^2
Barycentrics
P(117) = f(a, b, c) : f(b, c, a) : f(c, a, b)
U(117) = f(a, c, b) : f(b, a, c) : f(c, b, a)
where
f(a, b, c) =(b^2-c^2)*(a^4+K*a^2-b^2*c^2) + S*sqrt(2*((9*S^2+SW^2)*SW-K^3)*R^2-(S^2+SW^2)^2)
K = sqrt(-3*S^2+SW^2)
(César Lozada - Apr 8, 2024)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(b^2 - c^2)(a^4 - b^2c^2 + a^2Z) - ( - 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 (b^2 - c^2)*(a^4 - b^2*c^2 - a^2*Z) + (V - W)^(1/2) : :, where
Z = (a^4 + b^4 + c^4 - b^2*c^2 - c^2*a^2 - a^2*b^2)^(1/2)
V = 2*a^2*b^2*c^2*Z^3
W = b^6*c^6 + c^6*a^6 + a^6*b^6 - 3*a^4*b^4*c^4 - (b^4*c^4 + c^4*a^4 + a^4*b^4)*Z^2
Barycentrics (b^2-c^2)*(a^4-b^2*c^2-a^2*sqrt(-3*S^2+SW^2))+2*S*sqrt(2*R^2*(-3*S^2+SW^2)^(3/2)-(S^2+SW^2)^2+2*(9*S^2+SW^2)*SW*R^2) : : (César Lozada - Apr 8, 2024)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 (b^2 - c^2)(a^4 - b^2c^2 - a^2Z) - (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 (b^2 - c^2)*(a^4 - b^2*c^2 + a^2*Z) + (- V - W)^(1/2) : : , where
Z = (a^4 + b^4 + c^4 - b^2*c^2 - c^2*a^2 - a^2*b^2)^(1/2)
V = 2*a^2*b^2*c^2*Z^3
W = b^6*c^6 + c^6*a^6 + a^6*b^6 - 3*a^4*b^4*c^4 - (b^4*c^4 + c^4*a^4 + a^4*b^4)*Z^2
Barycentrics (b^2-c^2)*(a^4-b^2*c^2+a^2*sqrt(-3*S^2+SW^2))+2*S*sqrt(-2*R^2*(-3*S^2+SW^2)^(3/2)-(S^2+SW^2)^2+2*(9*S^2+SW^2)*SW*R^2) : : (César Lozada - Apr 8, 2024)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 (b^2 - c^2)(a^4 - b^2c^2 + a^2Z) - ( - V - W)^(1/2)
P(119) and U(119) lie on central line X(2)X(1340).
P(120) = 1st INTERCEPT OF LINE X(7)X(812) AND ADAMS CIRCLE
Barycentrics c*(a - c)*(a + b - c)*(a - b + c)*(a^2 + b^2 - a*c - b*c) : :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)(a^3 + c^3 - a^2b + b^2c - 2ac^2 - 2b*c^2 + 2a*b*c) : :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 (b*a^2)^(1/3) : (c*b^2)^(1/3) : (a*c^2)^(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 (a^3+a^2*b-a*b^2-b^3-a^2*c-b^2*c-a*c^2+b*c^2+c^3)/a : :
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^5*(c+a)+b^4*c*a-2*b^3*(c^3+a^3)+b*(c-a)^2*(c+a)*(c^2+a^2)-c*a*(c^2-a^2)^2)/a : :
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 (-3*b^4 + 2*b^2*(c^2 + a^2) + (c^2 - a^2)^2)/a : :
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 (3*b^2 - c^2 - a^2)/a : :
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)
Bicentric Pairs PU(133)-PU(145)
César Lozada, January 26, 2016It 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 b*c/(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).
Bicentric Pairs PU(146)-PU(159)
César Lozada, March 28, 2016Bicentric 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^ : :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)
P(160) = SIMILICENTER OF 1ST MONTESDEOCA BISECTOR TRIANGLE AND EXCENTRAL TRIANGLE
Trilinears a*(-a^2*c + a*b*(b + c) + b*c*(2*b + c) ): :
Barycentrics a^2*(-a^2*c + a*b*(b + c) + b*c*(2*b + 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)
See also X(14163).
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 (5*a^4-(b^2+c^2)*a^2-2*(b^2-c^2)*(2*b^2-2*c^2-3*S))/a : :
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 (-3*S^2+2*a^2*(b^2+c^2-a^2)-2*(b^2-c^2)*S)/a : :
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 (-10*S^2+9*a^2*(b^2+c^2-a^2) - 12*(b^2-c^2)*S)/a : :
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 (6*S^2+a^2*(b^2+c^2-a^2) - 4*(b^2-c^2)*S)/a : :
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 (16*S^2+a^2*(b^2+c^2-a^2) - 6*(b^2-c^2)*S)/a : :
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 b*c/(c^2 + a^2) : :
Barycentrics 1/(c^2 + a^2) : :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 (b^2*(c^4 + a^4 - b^4))/a : :
Barycentrics b^2*(c^4 + a^4 - b^4) : :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 b*c*(b^2/(c^2 - a^2)) : :
Barycentrics b^2/(c^2 - a^2) : :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(2*B)*sin(C - A)^2 : :
Barycentrics sin(2*B)*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) : :
Barycentrics -2*sqrt(3)*(b^2-c^2)*S+(b^2+c^2)*a^2-(b^2-c^2)^2 : : (César Lozada - Apr 12, 2024)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(3)*(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 - a^2*b^2 - a^2*c^2) - 2*sqrt(3)*(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(B)^2*(cos(2*C) - cos(2*A)) : :
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/a*b*c)[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).
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).
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;
ℋ1 = locus of H1 as M traverses BC;
ℋ2 = 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).
P(211) = 4TH MONTESDEOCA POINT
Barycentrics (a + b - 3*c)*(a + b - c) : :Contributed by Angel Montesdeoca, November 29, 2022
In the plane of a triangle ABC, let
A'B'C' = medial triangle of ABC;
(I) = incircle;
Ta = line, other than BC, through A' tangent to (I);
Ba = Ta∩AC, and define Cb and Ac cyclically;
Ca = Ta∩AB, and define Ab and Bc cyclically;
Then the triangles CbCaAb and A'B'C' are perspective; let P denote their perspector, and triangles CbAcBa and A'B'C' are perspective; let U denote their perspector. The points P and U are a bicentric pair.X(28818) = barycentric product P*U
X(4859) = bicentric sum of P and U
X(3667) = bicentric difference of P and U
X(3667) = ideal point of line PU
X(4859) = midpoint of P and U
P(212) = 5TH MONTESDEOCA POINT
Barycentrics a*((a - c)^2 - b^2)*(a^3*b*(b - c) - a^2*(b^3 - 4*b*c^2 + c^3) + a*c*(b^3 - 2*b^2*c + c^3) - b^2*(b - c)^2*c) : :Contributed by Angel Montesdeoca, January 29, 2023
In the plane of a triangle ABC, let
I = X(1) = incenter
A2 = the point, other than B, where line BC meets circle (ABI)
A3 = the point, other than C, where line BC meets circle (ACI)
Ab = BI∩AA3, and define Bc and Ca cyclically
Ac = BI∩AA2, and define Ba and Cb
P = center of perspeconic of ABC and BcCbAc
U = center of perspeconic of ABC and CaAbBc
Then P and U are a bicentric pair given by 1st barycentric as shown above.
P(213) = 1ST NONREAL FOCUS OF MACBEATH INCONIC
Barycentrics P(213) = a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4 - (2*I)*(b^2 - c^2)*S : :
Barycentrics P(213) = (-I + cot(B))*(I + cot(C)) : :The 2nd focus of the MacBeath inconic is U(213) = a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4 + (2*I)*(b^2 - c^2)*S : : = (I + cot(B])*(-I + cot(C]) : :
Contributed by Peter Moses, September 17, 2023
P(213) lies on this line: {5, 523}
P(213) and U(213) lie213) on the Yff hyperbola and the following curves: K005, K009, K027, K028, K072, K164, K165, K166, K187, K244, K433, K526, K527, K583, K759, K762, K799, K835, K846, K850, K932, K1154, K1155, K1180, K1282, K1318, K1319, K1320, Q064, Q162
P(213) = reflection of U(213) in X(5)
P(213) = isogonal conjugate of U(213)
P(213) = reflection of U(213) in the Euler line
P(213) =barycentric quotient X(6)/U(213)
Properties of P(213) and U(213) together:
X(5) = midpoint
X(5) = bicentric sum
X(523) = bicentric difference
X(49) = crosssum
X(50) = crossdifference
X(94) = trilinear pole of line PU
X(523) = ideal point of line PU
X(93) = cevapoint
X(49) = crosspoint
X(4) = vertex conjugate
X(1) = trilinear product
X(6) = barycentric product
P(214) = 1ST NONREAL FOCUS OF YFF HYPERBOLA
Barycentrics P(214) = (cot(A) - 2*I)*cot(B) + cot(C)*(2*I + cot(A) + 4*cot(B)) : :
The 2nd nonreal focus of the Yff hyperbola is U(214) = (cot(A) - 2*I)*cot(C) + cot(B)*(2*I + cot(A) + 4*cot(C)) : :The real foci of the Yff hyperbola are X(3) and X(3830).
Triangle centers associated with PU(214):
X(381) = midpoint
X(523) = ideal point
X(58265) = barycentric product
X(58266) = crosssum
X(58267) = crossdifference
X(58268) = trilinear pole
P(215) = INFINITY POINT OF 1ST ASYMPTOTE OF NEUBERG-GIBERT HYPERBOLA
Barycentrics P(215) = sqrt(3)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4) - 2*(b^2 - c^2)*S : :
The infinity point of the 2nd asymptote of the Neubert-Gibert hyperbola as is U(215) = sqrt(3)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4) + 2*(b^2 - c^2)*S : :
The Neuberg-Gibert hyperbola was introduced by Peter Moses in the preamble just before X(60603). P(215) and U(215) lies on the lline at infinity.
Triangle centers associated with PU(215):
X(2) = trilinear pole of line PU(215)
X(6) = crossdifference
X(30) = bicentric sum
X(523) = bicentric difference
X(10620) = crosssum
X(46423) = vertex conjugate
X(60738) = trilinear product
X(60739) = barycentric product
X(60740) = cevapoint
X(60741) = crosspoint
P(216) = ISOTOMIC CONJUGATE OF P(24)
Barycentrics P(216) = (b-c)*(c-a) : :
P(217) = ISOTOMIC CONJUGATE OF P(178)
Barycentrics P(217) = a^2*c^2/(c^2 - a^2) : :Triangle centers associated with PU(217):
X(661) = trilinear product
X(512) = barycentric product
X(1084) = bicentric sum
X(888) = bicentric difference
X(99) = crossdifference
X(512) = trilinear pole of line PU
X(888) = ideal point of line PU
X(1084) = midpoint
X(25054) = crosspoint
P(218) = ISOTOMIC CONJUGATE OF P(180)
Barycentrics P(2183) = 1/(2*b^2 - c^2 - a^2) : :Triangle centers associated with PU(218):
X(14210) = trilinear product
X(524) = barycentric product
X(2482) = bicentric sum
X(690) = bicentric difference
X(41404) = crosssum
X(111) = crossdifference
X(524) = trilinear pole of line PU
X(690) = ideal point of line PU
X(2482) = midpoint
X(8591) = crosspoint
X3455) = vertex conjugate
P(219) = 1ST ISOBARYC OF X(519)
Barycentrics P(219) = 2*b - c - a : :Triangle centers associated with PU(219):
X(20568) = trilinear product
X(903) = barycentric product
X(519) = bicentric sum
X(514) = bicentric difference
X(21781) = crosssum
X(6) = crossdifference
X(2) = trilinear pole of line PU
X(9460) = crosspoint
P(220) = ISOTOMIC CONJUGATE OF P(219)
Barycentrics P(220) = 1/(2*b - c - a) : :Triangle centers associated with PU(220):
X(4358) = trilinear product
X(519) = barycentric product
X(4370) = bicentric sum
X(900) = bicentric difference
X(41461) = crosssum
X(106) = crossdifference
X(519) = trilinear pole of line PU
X(900) = ideal point of line PU
X(4370) = midpoint
X17487() = crosspoint
P(221) = 1ST ISOBARYC OF X(824)
Barycentrics P(221) = c^3 - a^3 : :Triangle centers associated with PU(221):
X(789) = trilinear product
X(4586) = barycentric product
X(752) = bicentric sum
X(824) = bicentric difference
X(6) = crossdifference
X(2) = trilinear pole of line PU
P(222) = ISOGONAL CONJUGATE OF P(221)
Barycentrics P(222) = a^2/(c^3 - a^3) : :Triangle centers associated with PU(222):
X(788) = trilinear product
X(46386) = barycentric product
X(4586) = crossdifference
X(3250) = trilinear pole of line PU
P(223) = ISOTOMIC CONJUGATE OF P(221)
Barycentrics P(223) = 1/(c^3 - a^3) : :Triangle centers associated with PU(223):
X(824) = barycentric product
X(61065) = bicentric sum
X(33904) = bicentric difference
X(825) = crossdifference
X(824) = trilinear pole of line PU
X(33904) = ideal point of line PU
X(61065) = midpoint
X(39356) = crosspoint
P(224) = 1ST ISOBARYC OF X(826)
Barycentrics P(224) = c^4 - a^4 : :Triangle centers associated with PU(224):
X(4593) = trilinear product
X(4577) = barycentric product
X(754) = bicentric sum
X(826) = bicentric difference
X(6) = crossdifference
X(2) = trilinear pole of line PU
P(225) = ISOGONAL CONJUGATE OF P(224)
Barycentrics P(225) = a^2/(c^4 - a^4) : :Triangle centers associated with PU(225):
X(2084) = trilinear product
X(688) = barycentric product
X(4577) = crossdifference
X(3005) = trilinear pole of line PU
P(226) = ISOTOMIC CONJUGATE OF P(224)
Barycentrics P(226) = 1/(c^4 - a^4) : :Triangle centers associated with PU(226):
X(826) = barycentric product
X(15449) = bicentric sum
X(33907) = bicentric difference,33907 X(827) = crossdifference
X(826) = trilinear pole of line PU
X(33907) = ideal point of line PU
X(15449) = midpoint
X(39346) = crosspoint
P(227) = 1ST ISOBARYC OF X(192)
Barycentrics P(227) = c*a - a*b - b*c : :Triangle centers associated with PU(227):
X(6384) = trilinear product
X(330) = barycentric product
X(75) = bicentric sum
X(513) = bicentric difference
X(3550) = crosssum
X(21760) = crossdifference
X(32020) = trilinear pole of line PU
X(513) = ideal point of line PU
X(75) = midpoint
X(3551) = cevapoint
P(228) = ISOGONAL CONJUGATE OF P(227)
Barycentrics P(228) = a^2/(c*a - a*b - b*c) : :Triangle centers associated with PU(228):
X(2209) = trilinear product
X(43) = bicentric sum
X(8640) = bicentric difference
X(40881) = crossdifference
X(3550) = crosspoint
X(2162) = vertex conjugate
P(229) = ISOTOMIC CONJUGATE OF P(227)
Barycentrics P(229) = 1/(c*a - a*b - b*c) : :Triangle centers associated with PU(229):
X(6376) = trilinear product
X(192) = barycentric product
X(6376) = bicentric sum
X(3083) = bicentric difference
X(17105) = crosssum
X(51864) = crossdifference
X(40844) = trilinear pole of line PU
X(4083) = ideal point of line PU
X(6376) = midpoint
X(24524) = crosspoint
P(230) = 1ST ISOBARYC OF X(29017)
Barycentrics P(230) = a*b*c*(a - b) + a^4 - b^4 : :Triangle centers associated with PU(230):
X(29017) = bicentric difference
X(6) = crossdifference
X(2) = trilinear pole of line PU
P(231) = 1ST ISOBARYC OF X(62428)
Barycentrics P(231) = a*b*c*(b - a) + a^4 - b^4 : :Triangle centers associated with PU(231):
X(6) = crossdifference
X(2) = trilinear pole of line PU
P(232) = 1ST ANGELICA POINT
Barycentrics P(232) = -3*a + 2*b + c : :Triangle centers associated with PU(232):
X(52885) = barycentric product
X(519) = bicentric sum
X(514) = bicentric difference
X(6) = crossdifference
X(2) = trilinear pole of line PU
P(233) = 2ND ANGELICA POINT
Barycentrics P(233) = a + 2*b - 3*c : :Triangle centers associated with PU(233):
X(519) = bicentric sum
X(514) = bicentric difference
X(6) = crossdifference
X(2) = trilinear pole of line PU
P(234) = 1ST ALOE POINT
Barycentrics P(234) = c^2 - a^2 + c*a - a*b - b*c : :Triangle centers associated with PU(234):
X(17365) = bicentric sum
X(4977) = bicentric difference
X(4977) = ideal point of line PU
X(17365) = midpoint
P(235) = 2ND ALOE POINT
Barycentrics P(235) = c^2 - a^2 - c*a + a*b + b*c : :Triangle centers associated with PU(235):
X(17362) = bicentric sum
X(900) = bicentric difference
X(900) = ideal point of line PU
X(17362) = midpoint
P(236) = 3RD ANGELICA POINT
Barycentrics P(236) = -3*a^2 + 2*b^2 + c^2 : :Triangle centers associated with PU(236):
X(52886) = barycentric product
X(524) = bicentric sum
X(523) = bicentric difference
X(6) = crossdifference
X(2) = trilinear pole of line PU
P(237) = 4TH ANGELICA POINT
Barycentrics P(237) = a^2 + 2*b^2 - 3*c^2 : :Triangle centers associated with PU(237):
X(524) = bicentric sum
X(523) = bicentric difference
X(6) = crossdifference
X(2) = trilinear pole of line PU
P(238) = 1ST ISOBARYC OF X(360)
Barycentrics P(238) = B : C : ABarycentrics of U(238) are C : A : B
P(239) = 1ST IMAGINARY FOCUS OF STEINER INELLIPSE OF CIRCUM-MEDIAL TRIANGLE
Barycentrics i*(b^2 - c^2)*(a^2 + b^2 + c^2)*(a^4 + 5*a^2*b^2 - 2*b^4 + 5*a^2*c^2 + 2*b^2*c^2 - 2*c^4) + 2*(8*a^6 - 3*a^4*b^2 - 9*a^2*b^4 + 2*b^6 - 3*a^4*c^2 - 3*a^2*b^2*c^2 - 6*b^4*c^2 - 9*a^2*c^4 - 6*b^2*c^4 + 2*c^6)*S : :See Peter Moses, euclid 7034.
P(239) and U(239) lie on this line: {4108, 8704}
P(239) and U(239) are the 1st and 2nd imaginary foci of the Stiner inellipse of the circum-medial triangle.
Editions
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.
11th edition, to PU(213): 9/17/2023
12th edition, to PU237) : 4/02/2024
13th edition, PU(198-237): 4/15/2024, by César Lozada.
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Last update: 04/03/2024 11:11:52
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