X1512–X1600

Orthojoins, Moses Circle Centers & Star-Named Centers

Orthojoins H(X) — orthopoles of the trilinear polars of isogonal conjugates (X1512–X1568) — together with Moses circle similicenters and star-named centers (Capella, Caph, Cebalrai, Celaeno, Chertan, Chort).
< 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_1514 = ORTHOJOIN OF X(3) (0.6818, 1.1255) X_1515 = ORTHOJOIN OF X(4) (0.5725, -0.0213) X_1518 = ORTHOJOIN OF X(8) (0.7658, 1.0514) X_1519 = ORTHOJOIN OF X(9) (0.8299, 1.2622) X_1521 = ORTHOJOIN OF X(11) (0.5709, 0.8250) X_1523 = ORTHOJOIN OF X(14) (0.6538, 0.5898) X_1525 = ORTHOJOIN OF X(16) (0.4851, 0.6110) X_1527 = ORTHOJOIN OF X(18) (0.7725, 0.9481) X_1528 = ORTHOJOIN OF X(19) (0.5409, 0.1119) X_1531 = ORTHOJOIN OF X(32) (0.0445, -0.5419) X_1534 = ORTHOJOIN OF X(40) (0.6016, 0.9221) X_1542 = ORTHOJOIN OF X(56) (0.2560, -0.4283) X_1546 = ORTHOJOIN OF X(62) (0.0919, -0.4180) X_1550 = ORTHOJOIN OF X(110) (0.9211, 0.1083) X_1551 = ORTHOJOIN OF X(111) (0.6353, 0.4562) X_1552 = ORTHOJOIN OF X(112) (0.7167, 0.3558) A B C X3 X1 X1514 X1515 X1518 X1519 X1521 X1523 X1525 X1527 X1528 X1531 X1534 X1542 X1546 X1550 X1551 X1552 (a, b, c) = (5, 6, 7) | AB = 1

X1512–X1600 (Orthojoins, Moses Circle Centers & Star-Named Centers) — 16 representative centers (from 89 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Group Preamble — Orthojoins: X1512-X1568

The orthojoin of a point X =\(x : y : z\) other than X6 is defined in the Glossary algebraically in terms of variables a,b,c. When these are sidelengths of a triangle, orthojoin(X) is the orthopole of the trilinear polar of the isogonal conjugate of X. (Added to ETC 6/18/03.) Let

D(a,b,c) = bc[2abcx + \(c^{3}\)y + \(b^{3}\)z - bc(by + cz) - \(a^{2}(cy + bz)\)],
E(a,b,c) = [(\(a^{4}\) - (\(b^{2}\) - \(c^{2})\)2]x - \(4a^{2}\)bc(y cos B + z cos C),
\(f(a,b,c) = D(a\),b,c)E(a,b,c).

Then orthojoin(X) =\(f(a,b,c) : f(b,c,a) : f(c,a,b)\). Below, orthojoin(X) is written as H(X).

Suppose X is not X2 and does not lie on a sideline of triangle ABC. The crossdifference of X and X2 has first trilinear
a(b-c)x’, where x’ = (by - cz)/(b-c).
Let \(X ^{-1}\) denote the isogonal conjugate of X. Then
\(H(X ^{-1})\) = (by - cz) cos A : (cz - ax) cos B : (ax - by) cos C.

In other words, if X lies on a line PG through the centroid G, then \(H(X ^{-1})\) lies on the line HQ, where H denotes the orthocenter and Q is a point that can be determined from the above formula. Examples:

If X is on the Euler line, L(2,3), then \(H(X ^{-1})\) is on the line L(4,6);
If X is on L(2,6,), then \(H(X ^{-1})\) is on the Euler line;
If X is on L(1,2), then \(H(X ^{-1})\) is on L(4,9);
If X is on L(2,7), then \(H(X ^{-1})\) is on L(1,4).

Suppose P is not X6, and let

S = crossdifference(P, X6) (S lies on the line at infinity) S' = orthopoint <orthopoint>`(S) (:g:`S' lies on the line at infinity) S″ = complementary conjugate of S' (S″ lies on the nine-point circle)

Let X be a point on line PX6 and not on a sideline of ABC. Then \(H(X ^{-1})\) is on line S″S'. Examples:

If X is a center on L(1,6) and X is not X6, then \(H(X ^{-1})\) is on L(119,517).
If X is a center on L(2,6) and X is not X6, then \(H(X ^{-1})\) is on L(114,511).
If X is a center on L(3,6) and X is not X6, then \(H(X ^{-1})\) is on L(113,30).
If X is a center on L(6,31) and X is not X6, then \(H(X ^{-1})\) is on L(118,516).
If X is a center on L(6,44) and X is not X6, then \(H(X ^{-1})\) is on L(117,515).

Further,

H(X11) = L(117,515)∩L(118,516) H(X37) = L(117,515)∩L(119,517) H(X244) = L(118,516)∩L(119,517)

If X lies on the line L(230,231), then H(X) lies on the nine-point circle. Examples:

H(X230) = X114 H(X231) = X128 H(X232) = X132 H(X468) = X1560 H(X523) = X115 H(X647) = X125 H(X650) = X11

X_1515 — ORTHOJOIN OF X_4

X1515ORTHOJOIN OF X4

X1515 = intersection of Simson line of X107 (line X133X1515) and trilinear polar of X107 (line X4X6)
X_1529 — ORTHOJOIN OF X_25

X1529ORTHOJOIN OF X25

As a point on the Euler line, X1529 has Shinagawa coefficients \((FS^{2}, -(E + F)[4(E + F)F - S^{2}])\).

X_1532 — ORTHOJOIN OF X_37

X1532ORTHOJOIN OF X37

As a point on the Euler line, X1532 has Shinagawa coefficients \((\$a\$S^{2}, \$a\$S^{2} + 2abc(E + F) + 2\$a(S_{A})^{2}\$)\).

X_1551 — ORTHOJOIN OF X_111

X1551ORTHOJOIN OF X111

As a point on the Euler line, X1551 has Shinagawa coefficients \((2(E + F)^{3} - 9(E - 2F)S^{2}, -3(E + F)[(E +10F)(E + F) - 6S^{2}])\).

X_1556 — ORTHOJOIN OF X_251

X1556ORTHOJOIN OF X251

As a point on the Euler line, X1556 has Shinagawa coefficients \((2(E + F)^{3} + (E + 2F)S^{2}, -(E + F)[5(E + 6F)(E + F) - 2S^{2}])\).

X_1557 — ORTHOJOIN OF X_263

X1557ORTHOJOIN OF X263

As a point on the Euler line, X1557 has Shinagawa coefficients \(([(E + F)^{3}F + 2(E + F)^{2}S^{2} + S^{4}] S^{2}, -4(E + F)^{3}F + (E - 11F)(E + F)^{3}S^{2} - (E + 10F)(E + F)S^{4} - 3S^{6})\).

X_1563 — ORTHOJOIN OF X_588

X1563ORTHOJOIN OF X588

As a point on the Euler line, X1563 has Shinagawa coefficients \(([2(E + F)^{2} + (9E + 8F)S+10S^{2}]S, -(E + F)^{2}E - 2(5E - F)(E + F)S - 4(7E - 2F)S^{2} - 22S^{3})\).

X_1564 — ORTHOJOIN OF X_589

X1564ORTHOJOIN OF X589

As a point on the Euler line, X1564 has Shinagawa coefficients \(([2(E + F)^{2} - (9E + 8F)S+10S^{2}]S, (E + F)^{2}E - 2(5E - F)(E + F)S + 4(7E - 2F)S^{2} - 22S^{3})\).

X_1589 — POINT CHARA I

X1589POINT CHARA I

As a point on the Euler line, X1589 has Shinagawa coefficients \((F + S, -S)\).

X_1590 — POINT CHARA II

X1590POINT CHARA II

As a point on the Euler line, X1590 has Shinagawa coefficients \((F - S, S)\).

X_1594 — RIGBY-LALESCU ORTHOPOLE

X1594RIGBY-LALESCU ORTHOPOLE

Traian Lalescu (Trajan Lalesco) proved in “A Class of Remarcable Triangles,” Gazeta Matematica 20 (1915) 213 [in Romanian], that if triangles DEF and D'E'F' are inscribed in a circle and directed arclengths satisfyarc DD' + arc EE' + arc FF' = 0 mod 2π, then the Simson lines of D, E, F with respect to D',E',F' and the Simson lines of D',E',F' with respect to D, E, F concur in the midpoint X of the segment of the orthocenters of DEF and D'E'F'. Daniel Vacaretu considered triangles DEF and D'E'F' associated with left and right isoscelizers and inscribed in the sine-triple-angle circle. He obtained the second set of trilinears shown above for the midpoint X. (See also the bicentric pair PU(61).)