X1166–X1208

Saragossa Points

1st, 2nd, and 3rd Saragossa points (Q, Q', Q″) constructed from the cevian and circumcevian triangles of classical centers.
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) ( A R_b ): Euclid I.22 Guide Circle (A) ( B R_a ): Euclid I.22 Guide Circle (B) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) circumcircle: Circumcircle of △ABC incircle: Incircle of △ABC A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) R_b = Radius Point R_b (0.8571, 0.0000) R_a = Radius Point R_a (0.2857, 0.0000) C = Vertex C (0.6122, 0.5999) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_1 = INCENTER (0.5714, 0.2333) X_1166 = 1st SARAGOSSA POINT OF X(5) (0.6183, 0.2746) X_1167 = 1st SARAGOSSA POINT OF X(34) (0.5172, 0.1910) X_1168 = 1st SARAGOSSA POINT OF X(36) (1.0000, 0.0000) X_1172 = 1st SARAGOSSA POINT OF X(48) (0.6100, 0.3717) X_1176 = 1st SARAGOSSA POINT OF X(66) (0.5328, 0.1263) X_1177 = 1st SARAGOSSA POINT OF X(67) (0.7694, -1.0073) X_1179 = 1st SARAGOSSA POINT OF X(96) (0.6245, 0.4442) X_1180 = 2nd SARAGOSSA POINT OF X(2) (0.5852, 0.2361) X_1181 = 2nd SARAGOSSA POINT OF X(3) (0.5894, 0.1559) X_1192 = 2nd SARAGOSSA POINT OF X(64) (0.6404, 0.3340) X_1204 = 3rd SARAGOSSA POINT OF X(64) (0.6497, 0.1923) X_1205 = 3rd SARAGOSSA POINT OF X(67) (0.7329, -0.3693) A B C X3 X1 X1166 X1167 X1168 X1172 X1176 X1177 X1179 X1180 X1181 X1192 X1204 X1205 (a, b, c) = (5, 6, 7) | AB = 1

X1166–X1208 (Saragossa Points) — 12 representative centers (from 43 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Group Preamble — Saragossa Points 1166-1208 (Saragossa points (1st, 2nd, 3rd))

Let A'B'C' be the cevian triangle of a point P, and let A″, B″, C″ be the respective intersections of lines PA, PB, PC with the circumcircle of triangle ABC. Let

U = B'C″∩B″C' V = C'A″∩C″A' W = A'B″∩A″B'.

Lines AU, BV, CW concur in the 1st Saragossa point of P;
lines A'U, B'V, C'W concur in the 2nd Saragossa point of P;
lines A″U, B″V, C″W concur in the 3rd Saragossa point of P.
These concurrences were presented by Darij Grinberg (Hyacinthos #6531, February 14, 2003),
with coordinates as follows. Let P =\(x : y : z\) (trilinears), and abbreviate the 1st, 2nd, and 3rd Saragossa points
as Q, Q', Q″, respectively; then first trilinears are
for Q: \(f(a,b,c) = a/[x(bz + cy)]\),
for Q': \(f(a,b,c) = ax[(b^{2}z^{2} + c^{2}y^{2})x + xyzbc + ayz(bz + cy)]\),
for Q″: \(f(a,b,c) = ax[(b^{2}z^{2} + c^{2}y^{2})x + ayz(bz + cy)]\).

The name Saragossa refers to the king who proved Ceva’s theorem before Ceva did. See

J. B. Hogendijk, “Al-Mu’taman ibn Hud [bar over u], 11th century king of Saragossa and brilliant mathematician,” Historia Mathematica, 22 (1995) 1-18.

The points P, Q', Q″ are collinear.

The 1st Saragossa point of Xi is Xj for these (i,j):
(1,58) (2,251) (3,4) (4,54) (6,6) (19,284) (21,961) (24,847) (25,2) (28,943) (31,81)
(32,83) (51,288) (55,57) (56,1) (58,1126) (64,3) (84,947) (154,1073) (184,275)
(198,282) (512,249) (513,59) (667,1016) (939,937) (1036,959).
The 2nd Saragossa point of Xi is Xj for these (i,j): (1,386) (6,6)
The 3rd Saragossa point of Xi is Xj for these (i,j): (3,185) (4,389) (6,6)
X_1200 — 3rd SARAGOSSA POINT OF X_55

X12003rd SARAGOSSA POINT OF X55

X1200 is the {X55,X1190}-harmonic conjugate of X41. For a list of other harmonic conjugates, click Tables at the top of this page.