Many of the terms used in triangle geometry are presented in Eric Weisstein's MathWorld, including Darij Grinberg's contributions. You can access these definitions directly from LIST 1 or MathWorld's search box. Other terms and terms that appeared initially in ETC or this Glossary are defined in LIST 2.
altitude, anticomplementary triangle, barycentric coordinates, Brocard angle (omega, ω), circle, circumcircle, circumradius, collinear points, conic, ellipse, equilateral triangle, Euler line, Euler triangle, excentral triangle, harmonic conjugate, homothetic triangles, hyperbola, inverse of a point in a circle, isoscelizer, Kiepert hyperbola, Kiepert parabola, line, line at infinity, medial triangle, midpoint, nine-point circle, orthic triangle, orthocentric system, parabola, pedal triangle, perspective triangles, power of a point, radical center, radical axis, rectangular hyperbola, Simson line, tangential triangle, triangle, trilinear coordinates LIST 1
LIST 2
In this list, points and lines are given in trilinears. Trilinears (or barycentrics) are understood to define points, lines, circles, triangle centers, etc., and zero determinants are understood to define collinearity and concurrence, etc., so that triangle geometry, formally speaking, is much more general than the study of a single euclidean triangle. In the formal treatment, sometimes called transfigured triangle geometry, the symbols a,b,c are regarded as algebraic indeterminates, so that points, defined as functions of a,b,c, are not the usual points of a two-dimensional plane. When (a,b,c) are real numbers restricted by the "triangle inequalities" for sidelengths, the resulting geometry is traditional triangle geometry. The triangle inequalities are stated here:
a ≤ b + c, b ≤ c + a, c ≤ a + b. In favor of this greater generality, an algebraic definition in this Glossary sometimes precedes the traditional geometric construction-definition.
The words "defined cyclically" apply after an "A-object" has been defined: if the A-object is F(A,B,C), then the B-object is F(B,C,A), obtained by the cyclic permutation ABC → BCA, and the C-object is, likewise, F(C,A,B). Example: If A' is the point where lines AP and BC meet, and B' and C' are defined cyclically, then B' is where lines BP and CA meet, and C' is where lines CP and AB meet.
aleph conjugate Suppose P = p : q : r and U = u : v : w are points, neither lying on a sideline of ABC. The P-aleph conjugate of U is the point
h(p,q,r,u,v,w) : h(q,r,p,v,w,u) : h(r,p,q,w,u,v),
whereh(p,q,r,u,v,w) = - q2r2u2 + r2p2v2 + p2q2w2 + (vw + wu + uv)( - q2r2 + r2p2 + p2q2). Let g be the mapping given by
g(x : y : z) = -x/p + y/q + z/r : x/p - y/q + z/r : x/p + y/q - z/r. The P-aleph conjugate of U is "what you do to g(X) to make g(X -1) when f(X) = U." (Hyacinthos #4111, Oct. 11, 2001.)
anticevian triangle Let P be a point not on a sideline of ABC. Let A' = AP∩BC. Let A" be the harmonic conjugate of P with respect to A and A'. Define B" and C" cyclically. Triangle A"B"C" is the anticevian triangle of triangle ABC. (The lines AP, BP, CP are the cevians of P, and A'B'C', the cevian triangle of P.)
If P = p : q : r, then A" = - p : q : r, B" = p : - q : r, C" = p : q : - r. Examples of anticevian triangles
excentral; P = incenter = X(1) = 1 : 1 : 1anticomplement If points P and Q are collinear with the centroid G, then P is the anticomplement of Q if G trisects segment PQ and is closer to Q than to P. See also complement. (According to Court, p. 297, the term anticomplementary point dates from 1886.)
anticomplementary; P = centroid = X(2) = 1/a : 1/b : 1/c
tangential; P = symmedian point = X(6) = a : b : canticomplementary triangle Let L(A) be the line through vertex A parallel to side BC, and define L(B) and L(C) cyclically. Let A' = L(B)∩L(C), and define B' and C' cyclically. Then A'B'C' is the anticomplementary triangle of ABC. (Efforts to replace this name have encountered the fact that A' is the anticomplement of A.)
anticomplementary conjugate Suppose P = p : q : r and U = u : v : w are points, neither lying on a sideline of ABC. The P-anticomplementary conjugate of U is the point
h(a,b,c,p,q,r,u,v,w) : h(b,c,a,q,r,p,v,w,u) : h(c,a,b,r,p,q,w,u,v),
whereh(a,b,c,p,q,r,u,v,w) = [b2/q(au + cw) + c2/r(au + bv) - a2/p(bv + cw)]/a. Let f(X) be the anticomplement of X, given by
f(x : y : z) = (- ax + by + cz)/a : (ax - by + cz)/b : (ax + by - cz)/c, and let k be P-isoconjugation. The P-anticomplementary conjugate of U is "what you do to f(X) to make f(k(X)) when f(X) = U."
If P = incenter, then P-anticomplementary conjugate is shortened to "anticomplementary conjugate."
antimedial triangle A synonym for anticomplementary triangle, defined above.
beth conjugate Suppose P = p : q : r and U = u : v : w are points, neither lying on a sideline of ABC. The P-beth conjugate of U is the point
h(a,b,c,p,q,r,u,v,w) : h(b,c,a,q,r,p,v,w,u) : h(c,a,b,r,p,q,w,u,v),
whereh(a,b,c,p,q,r,u,v,w) = 2abcp(cos B + cos C)(ua'/p + vb'/q + wc'/r) - (a+b+c)a'b'c'u, where a', b', c' are - a + b + c, a - b + c, a + b - c, respectively. Let f by the mapping given by
f(x : y : z) = p(y + z) : q(z + x) : r(x + y), and let k be the mapping "reflection in the circumcenter." The P-beth conjugate of U is "what you do to f(X) to make
f(k(X)) when f(X) = U." (Hyacinthos #4146, Oct. 26, 2001.)bicentric points Suppose f(a,b,c) : f(b,c,a) : f(c,a,b) is a point that satisfies (1) in the definition of triangle center, but that |f(a,b,c)| ≠ |f(a,c,b)|. Then
f(a,c,b) : f(b,a,c) : f(c,b,a) and f(a,b,c) : f(b,c,a) : f(c,a,b) are bicentric points, together comprising a bicentric pair. Example: the Brocard points, c/b : a/c : b/a and b/c : c/a : a/b.
center See triangle center.
central triangle Suppose each of f(a,b,c) and g(a,b,c) is a center function (i.e., as in the definition of triangle center) or the zero function, and that one of these conditions holds:
(1) the degree of homogeneity of g equals that of f;Let
(2) f is the zero function and g is not the zero function;
(3) g is the zero function and f is not the zero function.A' = f(a,b,c) : g(b,c,a) : g(c,a,b)Then A'B'C' is the (f,g)-central triangle of type 1. A central triangle of type 1 is any triangle A'B'C' for which equations (1)-(3) hold for some choice of center functions f and g.
B' = g(a,b,c) : f(b,c,a) : g(c,a,b)
C' = g(a,b,c) : g(b,c,a) : f(c,a,b).Next, suppose f and g are as for type 1 except that g(a,b,c) ≠ g(a,c,b). Let
A' = f(a,b,c) : g(b,c,a) : g(c,b,a)Then A'B'C' is the (f,g)-central triangle of type 2. A central triangle of type 2 is any triangle A'B'C' for some choice of center function f and function g as described.
B' = g(a,c,b) : f(b,c,a) : g(c,a,b)
C' = g(a,b,c) : g(b,a,c) : f(c,a,b).Examples: ABC, the medial, orthic, excentral, anticomplementary, and tangential triangles are central of type 1; in fact, if X is a center, then the cevian and anticevian triangles of X are central of type 1. Any pedal triangle that is not also a cevian triangle is a central triangle of type 2.
Ceva conjugate Suppose P = p : q : r and U = u : v : w are points, neither lying on a sideline of ABC. The P-Ceva conjugate of U is the point
u(- qru + rpv + pqw) : v(qru - rpv + pqw) : w(qru + rpv - pqw), which is the perspector of the cevian triangle of P and the anticevian triangle of U. (See cevian nest and cross conjugate.)
Ceva conjugate and cevapoint are related thus: X = P©U if and only if P = cevapoint(U,X), and in this case, U = P©X.
Let f(x : y : z) = p(y + z) : q(z + x) : r(x + y). The P-Ceva conjugate of U is "what you do to f(X) to make f(X -1) when f(X) = U." (Hyacinthos #3966, Sept. 26, 2001.)
If you have The Geometer's Sketchpad, you can view CEVA CONJUGATE.
cevapoint Suppose P = p : q : r and U = u : v : w are distinct points, neither lying on a sideline of ABC. The cevapoint of P and U is the point
(pv + qu)(pw + ru) : (qw + rv)(qu + pv) : (ru + pw)(rv + qw). Let A"B"C" be the anticevian triangle of U. Let A' =PA"∩BC, and define B' and C' cyclically. The cevapoint of P and U is the perspector, X, of triangles ABC and A'B'C'. Moreover, P is the X-Ceva conjugate of U, and U is the X-Ceva conjugate of P. (See cevian nest and Ceva conjugate.) As a comparison of cevapoint(P,U) and crosspoint(P,U), note that their trilinears can be written as
1/(qw + rv) : 1/(ru + pw) : 1/(pv + qu) and 1/qw + 1/rv : 1/ru + 1/pw : 1/pv + 1/qu. If you have The Geometer's Sketchpad, you can view CEVAPOINT.
Floor Van Lamoen contributed the following two constructions of cevapoint(P,U). Let ApBpCp be the cevian triangle of P, and let AuBuCu be the cevian triangle of U. Define
Then triangle ABC is perspective to both triangles ApuBpuCpu and AupBupCup, and the perspector in both cases is the cevapoint of P and U. (Received 10/17/03)
Apu = CpAu∩ApBu . . . . . . . . . . . . . . Aup = CuAp∩AuBp Bpu = ApBu∩BpCu . . . . . . . . . . . . . . Bup = AuBp∩BuCp Cpu = BpCu∩CpAu . . . . . . . . . . . . . . Cup = BuCp∩CuAp cevian Let P be a point not on a sideline of ABC. The lines AP, BP, CP are the cevians of P.
cevian nest Suppose D,E,F are triangles, and that F is inscribed in E, and E is inscribed in D. If any two of the three triangles are perspective, then the ordered triple (D,E,F) is a cevian nest. (It is well known that if any two of the three triangles are perspective, then each is perspective to the third; in particular, E is a cevian triangle of D, and F is a cevian triangle of E.)
Cevian nests in which one of the triangles is ABC beget three families of conjugates: Ceva conjugates, cross conjugates, and isoconjugates. See also crosspoint.)
If you have The Geometer's Sketchpad, you can view CEVIAN NEST.
cevian triangle Let P be a point not on a sideline of ABC. Let A' = AP∩BC meet. Define B' and C' cyclically. Triangle A'B'C' is the cevian triangle of triangle ABC.
If P = p : q : r, then A' = 0 : q : r, B' = p : 0 : r, C' = p : q : 0. Examples of cevian triangles:
incentral triangle; P = incenter = X(1) = 1 : 1 : 1cocevian triangle Let P be a point not on a sideline of ABC, and let A'B'C' be the anticevian triangle of P. Let A" be the harmonic conjugate of A' with respect to B and C, and define B" and C" cyclically. (If P = p : q : r, then A" = 0 : y : - z.). The cocevian triangle of P is the triangle A"B"C". The vertices are collinear and the triangle degenerate. For a discussion, see TCCT, page 200.
medial triangle; P = centroid = X(2) = 1/a : 1/b : 1/c
orthic triangle; P = orthocenter = X(4) = sec A : sec B : sec C
intouch triangle; P = Gergonne point = X(7) = sec2(A/2) : sec2(B/2) : sec2(C/2)
extouch triangle; P = Nagel point = X(8) = csc2(A/2) : csc2(B/2) : csc2(C/2)
complement If points P and U are collinear with the centroid G, then P is the complement of U if G trisects segment PU and is closer to P than to U. See also anticomplement. (According to Court, p. 297, the term complementary point dates from 1885.)
complementary conjugate Suppose P = p : q : r and U = u : v : w are points, neither lying on a sideline of ABC. The P-complementary conjugate of U is the point
h(a,b,c,p,q,r,u,v,w) : h(b,c,a,q,r,p,v,w,u) : h(c,a,b,r,p,q,w,u,v),
whereh(a,b,c,p,q,r,u,v,w) = [b2/q(au - bv + cw) + c2/r(au + bv - cw)]/a. Let f(X) be the complement of X, given by
f(x : y : z) = (bv + cw)/a : (cw + au)/b : (au + bv)/c, and let k be P-isoconjugation. The P-complementary conjugate of U is "what you do to f(X) to make f(k(X)) when f(X) = U."
If P = incenter, then P-complementary conjugate is shortened to "complementary conjugate."
cross conjugate Suppose P = p : q : r and U = u : v : w are distinct points, neither lying on a sideline of ABC. The P-cross conjugate of U is the point
u/(- pvw + qwu + ruv) : v/(pvw - qwu + ruv) : w/(pvw + qwu - ruv). Let A'B'C' be the cevian triangle of U. Let A" be where line PA' crosses line B'C', and define B" and C" cyclically - so that A"B"C" is the cevian triangle (in triangle A'B'C', not ABC) of P. The perspector of triangles ABC and A"B"C" is the P-cross conjugate of U. (See cevian nest.)
As a transformation, P-cross conjugate is an involution; i.e., if PxU = P-cross conjugate of U, then Px(PxU) = U. Properties of cross conjugates arise from those of Ceva conjugates; for, on writing PcU for the P-Ceva conjugate of U,
PxU = (U/P)*(UcP) and PcU = (U/P)*(UxP) where / and * denote trilinear division and multiplication.Cross conjugate and crosspoint are related thus: X = PxU if and only if U = crosspoint(X,P), and in this case, X = crosspoint(U,P).
If you have The Geometer's Sketchpad, you can view CROSS CONJUGATE.
crossdifference Suppose P = p : q : r and U = u : v : w are distinct points, neither lying on a sideline of ABC. The crossdifference of P and U is the point X defined by trilinears
qw - rv : ru - pw : pv - qu, constructible as the isogonal conjugate of the trilinear pole of the line PU. Thus, U is the crossdifference of P and X, and P is the crossdifference of U and X. In order to see that the crossdifference of P and X is a solution for U of the equation
x : y : z = qw - rv : ru - pw : pv - qu,
note first that this equation implies px + qy + rz = 0. Thus, if u : v : w = qz - ry : rx - pz : py - qx, then
qw - rv = q(py - qx) - r(rx - pz) = - (p2 + q2 + r2)x, so that
qw - rv : ru - pw : pv - qu = x : y : z.A point X is the crossdifference of distinct points P and U if and only if the line PU is the trilinear polar of the isogonal conjugate of X. Examples:
X(1) = crossdifference of X(44) and X(513)
X(2) = crossdifference of X(187) and X(512)
X(3) = crossdifference of X(230) and X(231).crosspoint Suppose P = p : q : r and U = u : v : w are distinct points, neither lying on a sideline of ABC. The crosspoint of P and U is the point
pu(rv + qw) : qv(pw + ru) : rw(qu + pv). This point is introduced in TCCT with the notation C(P,Q). Here, "Q" is replaced by "U", and C(P,U) is abbreviated as X. The construction for X given in TCCT follows. Let A'B'C' be the cevian triangle of P and A"B"C" that of U. Let A''' = AA"∩B'C', and define B''' and C''' cyclically. Then X is the perspector of triangles A'B'C' and A'''B'''C'''. Moreover, U is the X-cross conjugate of P, and P is the X-cross conjugate of U. (See cevian nest and cross conjugate.) The following properties of X were noted by Jean-Pierre Ehrmann (5/29/03):
X = perspector of A'B'C' and the triangle with vertices AU∩B'C', BU∩C'A', CU∩A'B';
X = perspector of A"B"C" and the triangle with vertices AP∩B"C", BP∩C"A", CP∩A"B";
X = the point of concurrence of these three lines:
the line of points AP∩BU and AU∩BP
the line of points BP∩CU and BU∩CP
the line of points CP∩AU and CU∩APFigure 7.3 in TCCT with "Q" replaced by "U" can be used to visualize the nine lines concurring in X.
If you have The Geometer's Sketchpad, you can view CROSSPOINT.
crosssum Suppose P = p : q : r and U = u : v : w are distinct points, neither lying on a sideline of ABC. The crosssum of P and U is the point X defined by trilinears
qw + rv : ru + pw : pv + qu, constructible as the crosspoint of the isogonal conjugates of P and U. Thus, U is the isogonal conjugate of the X-cross conjugate of the isogonal conjugate of P, and P is the isogonal conjugate of the X-cross conjugate of the isogonal conjugate of U. (Regarding the neologism "crosssum" placed here on 5/28/03, what words in the English language have spellings containing three consecutive identical letters?)
cubic A curve of degree 3 in barycentric or trilinear coordinates. For a catalog with sketches, visit Bernard Gibert's site.
cyclocevian conjugate Suppose P = p : q : r is a point not on a sideline of ABC, and A'B'C' is the cevian triangle of P. The circumcircle of A'B'C' meets line BC in two points: A' and A"; pairs B', B", and C',C" are obtained cyclically. The lines AA", BB", CC" concur in the cyclocevian conjugate of P. Let
g(a,b,c) = a/[p(qb + rc)] and f(a,b,c) = bc/[g(b,c,a) + g(c,a,b) - g(a,b,c)]. The cyclocevian conjugate of P is given by
f(a,b,c) : f(b,c,a) : f(c,a,b). In Hyacinthos #6423 (January 24, 2003), Darij Grinberg states nine theorems, of which the last is as follows:
The cyclocevian conjugate of a point is the
isotomic conjugate
of the anticomplement
of the isogonal conjugate
of the complement
of the isotomic conjugate
of the point.If you have The Geometer's Sketchpad, you can view CYCLOCEVIAN CONJUGATE.
eigencenter Suppose T is a central triangle and U(T) is its unary cofactor triangle. Then T and U(T) are perspective, and their perspector is the eigencenter of T.
Suppose the A-, B-, C- vertices of T are Pi = xi : yi : zi, for i = 1,2,3. Let
s = y3(x1x2 + z1z2) - y1(x2x3 + z2z3) Let x = st - uv, and define y and z cyclically. Then the eigencenter of T is the point x : y : z.t = z1(x2x3 + y2y3) - z2(x1x3 + y1y3)
u = z3(x1x2 + y1y2) - z1(x2x3 + y2y3)
v = y1(x2x3 + z2z3) - y2(x1x3 + z1z3)
gimel conjugate Suppose P = p : q : r and U = u : v : w are points, neither lying on a sideline of ABC. The P-gimel conjugate of U is the point
h(a,b,c,p,q,r,u,v,w) : h(b,c,a,q,r,p,v,w,u) : h(c,a,b,r,p,q,w,u,v),
whereh(a,b,c,p,q,r,u,v,w) = 2abc[- (cos A)/p + (cos B)/q + (cos C)/r]S - 8uσ2, where S = (bq + cr)u + (cr + ap)v + (ap + bq)w and σ = area of ABC.
Details on gimel conjugates, as well as aleph, beth, daleth, he, waw, zayin, complementary, and anticomplementary conjugates are given in C. Kimberling, "Collineations, Conjugacies, and Cubics," Forum Geometricorum 2 (2002) 21-32.
Hirst inverse Suppose P = p : q : r and U = u : v : w are distinct points, neither lying on a sideline of ABC. The P-Hirst inverse of U is the point
qru2 - vwp2 : rpv2 - wuq2 : pqw2 - uvr2. Geometrically, this is the point of intersection of the line PU and the polar of U with respect to the conic.
pyz + qzx + rxy = 0. As a transformation, P-Hirst inverse is an involution; i.e., if PhU = P-Hirst inverse of U, then Ph(PhU) = U. Concerning the designation "Hirst inverse," see the information contributed by Gunter Weiss.
isoconjugate Suppose P = p : q : r and U = u : v : w are points, neither on a sideline of ABC. The P-isoconjugate of U is the point
qrvw : rpwu : pquv. Examples: the isogonal conjugate of U is the X(1)-isoconjugate of U, and the isotomic conjugate of U is the X(31)-isoconjugate of U.
As suggested by the meaning of the prefix "iso",
P-isoconjugate of U = U-isoconjugate of P. Let A'B'C' be the anticevian triangle of U, and let A"B"C" be the anticevian triangle of P with respect to A'B'C'. The perspector of A'B'C' and A"B"C" is the U -2-isoconjugate of P, where U -2 = u -2 : v -2: w -2. (See cevian nest.)
The earliest appearance of the term "isoconjugate" in triangle geometry may its inclusion in this Glossary as early as 1998. Isoconjugates are also discussed in C. Kimberling, "Conjugacies in the plane of a triangle," Aequationes Mathematicae 63 (2002) 158-167, submitted April 27, 1999, and "Conics associated with a cevian nest," Forum Geometricorum 1 (2001) 141-150.
Thereafter the term "isoconjugate" was sometimes used for reciprocal conjugate, a term defined in this Glossary, with a reference to probable earliest usage.
isogonal conjugate Suppose P = p : q : r is a point not on a sideline of ABC. Let L(A) be the line obtained by reflecting line AP in the internal bisector of angle A. Define L(B) and L(B) cyclically. The lines L(A), L(B), L(C) concur in the isogonal conjugate of P, which has trilinears 1/p : 1/q : 1/r and is denoted by P -1. See isoconjugate.
If you have The Geometer's Sketchpad, you can view ISOGONAL CONJUGATE and ISOGONAL CONJUGATE OF A LINE.
isoscelizer An isoscelizer is a line perpendicular to an angle bisector. If P is a point, then the A-isoscelizer of P is the line L(P,A) through P perpendicular to the line that bisects vertex angle A; the B- and C- isoscelizers are defined cyclically. Let D and E be the points where L(P,A) meets sidelines AB and AC. Unless D = E = A, the triangle ADE is isosceles.
In ETC, there are several triangle centers defined in terms of isoscelizers. These were discovered or invented by Peter Yff, in whose notebooks the word isoscelizer dates back to 1963.
isotomic conjugate Suppose P = p : q : r is a point not on a sideline of ABC. Let A', B', C' be the points where lines AP, BP, CP meet lines BC, CA, AB, respectively. Reflect A', B', C' about the midpoints of sides BC, CA, AB, respectively, to obtain points A", B", C". Then lines AA", BB", CC" concur in the isotomic conjugate of P, which has trilinears 1/(pa)2 : 1/(qb)2 : 1/(rc)2. See isoconjugate.
line A line is the set of points x : y : z satisfying the equation
f(a,b,c)x + g(a,b,c)y + h(a,b,c)z = 0 for some point f(a,b,c) : g(a,b,c) : h(a,b,c). If this point is a triangle center, then the corresponding line is a central line.
Three lines dx + ey + fz = 0, rx + sy + tz = 0, ux + vy + wz = 0 are concurrent if and only the following determinant equation holds:
See also point. Chapter 5 of TCCT discusses central lines. line conjugate Suppose P = p : q : r and U = u : v : w are distinct points, neither equal to A, B, or C. The P-line conjugate of U is the point
p(v2 + w2) - u(qv + rw) : q(w2 + u2) - v(rw + pu) : r(u2 + v2) - w(pu + qv), or, equivalently,
hp - ku : hq - kv : hr - kw, where h = u2 + v2 + w2 and k = pu + qv + rw.
This is the point of intersection of line PU and the trilinear polar of the isogonal conjugate of U.
line-polar triangle Suppose Pi = xi : yi : zi are noncollinear points, for i = 1,2,3. An equation for line P2P3 is then
a1x + b1y + c1z = 0, where a1 = y2z3 - z2y3, and b1 and c1 are defined cyclically. The pole of this line is the point1/a1 : 1/b1 : 1/c1. This point is the A-vertex of the line-polar triangle. The B- and C- vertices are defined cyclically.major center a triangle center X for which there exists a function f(A) such that X = f(A) : f(B) : f(C). Examples: X(1), X(2), X(3), X(4), X(6). Major centers solve certain problems in functional equations; click here (AeqMath) or here (AMM) for more.
Consider two examples, X(9) and X(37), of which first trilinears are b + c - a and b + c, respectively. It is not clear from these trilinears that X(9) is a major center, whereas X(37) is not. Indeed, X(9) also has first trilinear cot(A/2), so that X(9) is a major center, but there remains this problem: how to establish that X(37) and others are not major. In April, 2008, Manol Iliev found a criterion for a triangle center to be not a major center. He applied his test to the first 3236 triangle centers in ETC and found that exactly 292 of them are major, as listed here:
1 2 3 4 6 7 8 9 13 14 15 16 17 18 19 24 25 31 32 33 34 35 36 41 47 48 49 50 55 56 57 61 62 63 68 69 75 76 77 78 79 80 85 91 92 93 94 158 173 174 179 184 186 188 200 202 203 212 215 219 220 222 236 255 258 259 264 265 266 269 273 278 279 281 289 298 299 300 301 302 303 304 305 312 317 318 319 320 323 326 328 331 340 341 345 346 348 357 358 359 360 365 366 371 372 378 393 394 400 470 471 472 473 479 480 483 485 486 491 492 506 507 508 509 554 555 556 557 558 559 560 561 562 563 571 577 601 602 603 604 605 606 607 608 728 738 847 999 1000 1028 1049 1077 1081 1082 1085 1088 1092 1093 1094 1095 1096 1102 1106 1115 1118 1119 1123 1124 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1139 1140 1143 1147 1151 1152 1250 1251 1253 1259 1260 1264 1265 1267 1270 1271 1274 1321 1322 1327 1328 1335 1336 1395 1397 1398 1399 1407 1411 1435 1442 1443 1488 1489 1496 1497 1501 1502 1583 1584 1585 1586 1593 1597 1598 1599 1600 1659 1748 1802 1804 1807 1820 1847 1857 1870 1917 1928 1969 1973 1974 1989 1993 1994 2003 2006 2052 2066 2067 2089 2151 2152 2153 2154 2160 2161 2165 2166 2174 2175 2207 2212 2289 2306 2307 2323 2351 2361 2362 2477 2671 2672 2673 2674 2962 2963 2964 2965 3043 3076 3077 3082 3083 3084 3092 3093 3179 3200 3201 3205 3206 3218 3219 medial triangle the cevian triangle of the centroid, X(2). The A-vertex of the medial triangle is the point A' in which the line A-to-X(2) meets the sideline BC. Vertices B' and C' are defined cyclically. (See cevian triangle.)
orthojoin Suppose X = x : y : z is a point, and let
a1 = (b2 + c2 - a2)/(2bc) = cos A,
b1 = (c2 + a2 - b2)/(2ca) = cos B,
c1 = (a2 + b2 - c2)/(2ab) = cos C,u1 = b1c1x3 - b1y3 - c1z3,
u2 = - (1 + 2c12)b1x2y - (1 + 2b12)c1x2z,
u3 = (2a1b1 - c1)y2z + (2a1c1 - b1)yz2,
u4 = 3b1c1xy2 + 3b1c1xz2,
u5 = (1 + b12 + c12 - a12 - 4a1b1c1)xyz,f(a,b,c) = (- x + yc1 + zb1)(u1 + u2 + u3 + u4 + u5).
The orthojoin of X is the point f(a,b,c) : f(b,c,a) : f(c,a,b).
The above formula for f(a,b,c) can be written out explicitly in terms of a,b,c and simplified to the form
g(a,b,c) : g(b,c,a) : g(c,a,b), whereg(a,b,c) = bc[2abcx + c3y + b3z - (a2 + bc)(cy + bz)]F,
where F = x[a4 - (b2 - c2)2] + 2a[by(b2 - a2 - c2) + cz(c2 - b2 - a2)]
The orthojoin of X, defined (6/16/03) as above, can be described geometrically (when a,b,c are sidelengths of a triangle) as the orthopole of the trilinear polar of the isogonal conjugate of X. The orthopole of a line L is constructed as follows. Let A' be the foot of the perpendicular from vertex A on L, and define B' and C' cyclically. Then the perpendiculars from A' to BC, from B' to CA, and from C' to AB concur in the orthopole of L. For a proof and link to further information, see Orthopole - a new proof by Darij Grinberg.
orthopoint Each point X on the line at infinity is the point common to every line in a family F(X) of pairwise parallel lines. That is, X may be regarded as the direction common to the lines in F(X). Let L be any line in F(X), and let L' be any line perpendicular to L. Let G be the family of lines parallel to L'. The orthopoint of X is the point H(X) in which L' meets the line at infinity. If X = x : y : z, then
H(X) = cy cos B - bz cos C : az cos C - cx cos A : bx cos A - ay cos B. For any X on the line at infinity, the isogonal conjugates of X and H(X) are circumcircle-antipodes; i.e., each is the X(3)-reflection of the other.perspective Triangles DEF and UVW are perspective triangles if one of the following six triples of lines concur in a point:
{DU,EV,FW}, {DV,EW,FU}, {DW,EU,FV}; {DU,EW,FV}, {DV,EU,FW}, {DW,EV,FU} The point of concurrence is called a perspector (replacing center of perspective). If, for example, lines DU,EV,FW concur in a point, then by Desargues's theorem, the points
EF intersect VW, FD intersect WU, DE intersect UV are collinear, and their line is the perspectrix (replacing axis of perspective.)
In case DEF is the reference triangle, ABC, and UVW is a central triangle, it is sometimes tacitly understood that perspectivity of the two triangles refers to only one of the six possibilities, namely, the concurrence of lines DU,EV,FW. In this case, if UVW is the cevian triangle, A'B'C', of a point P, then P is the perspector of ABC and A'B'C', and the perspectrix is the trilinear polar of P.
perspector (See perspective.)
perspectrix (See perspective.)
point In euclidean geometry, point is not subject to definition, much as an axiom is not subject to proof. However, when (a,b,c) is variable, a point is defined informally by f(a,b,c) : g(a,b,c) : h(a,b,c) for all (a,b,c) for which at least one of the values f(a,b,c), g(a,b,c), h(a,b,c) is not zero. A formal definition is given in TCCT. See also triangle center.
Three points x : y : z, r : s : t, u : v : w are collinear if and only the following determinant equation holds:
Thus, if x : y : z is a variable point, this equation gives the line determined by the points r : s : t and u : v : w. point-polar triangle Suppose Pi = xi : yi : zi are noncollinear points, none lying on a sideline of ABC, for i = 1,2,3. The polar of P1 is the line
x/x1 + y/y1 + z/z1 = 0, and the polars of P2 and P3 are defined cyclically. These last two lines meet in the point1/y2z3 - 1/z2y3 : 1/z2x3 - 1/x2z3 : 1/x2y3 - 1/y2x3. This point is the A-vertex of the point-polar triangle. The B- and C- vertices are defined cyclically.polar (trilinear polar) The trilinear polar of a point p : q : r is the line
qrx + rpy + pqz = 0. polar (of a point P with respect to a conic) Suppose W is a conic, P a point, and XX' and YY' chords of W that meet in P. As X and Y vary, the locus of the point X'Y∩XY' is a line, called the polar of P with respect to W. If W is a circumconic, it has an equation uyz + vzx + wxy = 0, and the polar of a point Q = p : q : r is the line
(vr + wq)x + (wp + ur)y + (uq + vp)z = 0. pole (trilinear pole) The trilinear pole of the line ux + vy + wz = 0 is the point wv : uw : uv.
polynomial center A triangle center X is a polynomial center if there exists a polynomial f(a,b,c) such that
X = f(a,b,c) : f(b,c,a) : f(c,a,b). Examples: X(1), X(2), . . . , X(12), but not X(13).
radical trace The radical trace of two nonconcentric circles is the point of intersection of the radical axis of the circles and the line of the centers of the circles. (For examples, see X(I) for I = 6, 187, 1570, 2021-2025, 2030-2032.)
reciprocal conjugate Suppose P and U are points, neither on a sideline of ABC, given in barycentric coordinates by P = p : q : r and U = u : v : w. The P-reciprocal conjugate of U is the point
pvw : qwu : ruv. (barycentrics) As suggested by the meaning of "reciprocal",
P-reciprocal conjugate of U = G/(U-reciprocal conjugate of P), in accord with the fact that G, the centroid, is the identity corresponding to barycentric division.
The term reciprocal conjugate was introduced in Keith Dean and Floor van Lamoen, "Geometric construction of reciprocal conjugations," Forum Geometricorum 1 (2001) 115-120.
See also isoconjugate.
symbolic substitution Suppose p(a, b, c), q(a, b, c), r(a, b, c) are functions of a, b, c, all of the same degree of homogeneity. As the transfigured plane consists of all points - that is, functions of the form X = x(a, b, c) : y(a, b, c) : z(a, b, c), the substitution indicated by
a ---> p(a, b, c), b--->q(a, b, c), c--->r(a, b, c) maps the set of all points - that is, the transfigured plane - into itself. Such a substitution has no clear geometric meaning, as suggested by the name, symbolic substitution. On the other hand, symbolic substitutions are of geometric interest because they map lines to lines, conics to conics, cubics to cubics, and they preserve incidence.Example: The symbolic substitution (a, b, c) --->(1/a, 1/b, 1/c) maps every triangle center to a triangle center, every pair of bicentric points to a pair of bicentric points, every circumconic to a circumconic, etc. However, when (a, b, c) = (2, 4, 5), for example, then a, b, c are sidelengths of a euclidean triangle, but 1/a, 1/b, 1/c are not.
Symbolic substitutions are introduced in C. Kimberling, "Symbolic substitutions in the transfigured plane of a triangle," Aequationes Mathematicae 73 (2007) 156-171.
transcendental center A triangle center X is a transcendental center if there exists no algebraic function f(a,b,c) such that X = f(a,b,c) : f(b,c,a) : f(c,a,b). Examples: X(359) and X(360).
triangle center A triangle center is a point of the form f(a,b,c) : f(b,c,a) : f(c,a,b), where f is a nonzero function satisfying two conditions:
(1) f is homogeneous in a,b,c; i.e., there is a nonnegative real number h such thatA formal definition of triangle center is given in TCCT. See also polynomial center, transcendental center, central triangle, bicentric points.f(ta,tb,tc) = thf(a,b,c) for all (a,b,c) in the domain of f; (2) f is symmetric in b and c; i.e., f(a,c,b) = f(a,b,c).
unary cofactor triangle Suppose Pi = xi : yi : zi are points, for i = 1,2,3. The A-vertex of the unary cofactor triangle is the point
y2z3 - z2y3 : z2x3 - x2z3 : x2y3 - y2x3 . The B-vertex and C-vertex are defined cyclically.The vertices are the isogonal conjugates of the the vertices of the line-polar triangle of the points Pi. If T is a triangle and U(T) its unary cofactor triangle, then U(U(T)) = T, and T and U(T) are perspective; see eigencenter.
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