X3218–X3309

Side-Swapping SS(a → bc) Points, Parabola Vertices & Morley Equilateral Triangles

Side-swapping transformations SS(a → bc), vertices of the inscribed Kiepert and Yff parabolas (X3233–X3234), reflections in X1 (X3241–X3245), and centers and perspectors of the six equilateral triangles related to the 1st, 2nd, and 3rd Morley triangles (X3272–X3283).
< 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_3219 = INTERSECTION OF LINES X(2)X(7) AND X(8)X(90) (0.4976, 0.1420) X_3221 = SS(a → bc) OF X(647) (0.9096, -0.2538) X_3225 = SS(a → bc) OF X(98) (1.1195, 0.2435) X_3226 = SS(a → bc) OF X(105) (1.0976, 0.1415) X_3233 = VERTEX OF THE INSCRIBED KIEPERT PARABOLA (0.4343, 0.4780) X_3235 = INTERSECTION OF LINES X(1)X(32) and X(3)X(2007) (0.5954, 0.2683) X_3248 = CROSSSUM OF X(1) AND X(190) (0.8063, 0.1545) X_3249 = SS(a → bc) OF X(764) (1.2119, -0.1962) X_3251 = INTERSECTION OF X(1)X(513) AND X(42)X(663) (0.3571, 0.3499) X_3258 = CROSSSUM OF X(74) AND X(110) (0.3149, 0.1657) X_3276 = CENTER OF 2nd MORLEY TRIANGLE (0.4192, -0.0726) X_3284 = INTERSECTION OF LINES X(2)X(340) AND X(3)X(6) (0.1903, -0.4095) X_3285 = INTERSECTION OF LINES X(3)X(6) AND X(21)X(45) (1.0773, 1.0556) X_3289 = INTERSECTION OF LINES X(2)X(6) AND X(3)X(217) (-0.0220, -0.4012) X_3290 = INTERSECTION OF LINES X(2)X(37) AND X(230)X(231) (1.0664, 0.6348) X_3292 = INTERSECTION OF LINES X(3)X(49) AND X(23)X(110) (-0.3161, -0.3898) A B C X3 X1 X3219 X3221 X3225 X3226 X3233 X3235 X3248 X3249 X3251 X3258 X3276 X3284 X3285 X3289 X3290 X3292 (a, b, c) = (5, 6, 7) | AB = 1

X3218–X3309 (Side-Swapping SS(a → bc) Points, Parabola Vertices & Morley Equilateral Triangles) — 16 representative centers (from 92 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Group Preamble — Points Associated with Equilateral Triangles (are associated with six equilateral triangles)

The points X3272 to X3283 are associated with six equilateral triangles related to the (1st) Morley triangle; trilinears for vertices of these triangles are given just below. Received from Milorad R. Stevanovic, Nov. 24, 2007 and Dec. 25, 2007.

1st Morley triangle, M1M2M3, where
M1 = 1 : 2 cos C/3 : 2 cos B/3
M2 = 2 cos \(C/3 : 1 : 2\) cos A/3
M3 = 2 cos B/3 : 2 cos A/3 : 1
2nd Morley triangle, P1P2P3, where
P1 = 1 : 2 cos(C/3 - 2π/3) : 2 cos(B/3 - 2π/3)
P2 = 2 \(\cos(C/3 - 2\pi /3) : 1 : 2\) cos(A/3 - 2π/3)
P3 = 2 cos(B/3 - 2π/3) : 2 cos(A/3 - 2π/3) : 1
3rd Morley triangle, \(S_{1}\)\(S_{2}\)\(S_{3}\), where
\(S_{1}\) = 1 : 2 cos(C/3 - 4π/3) : 2 cos(B/3 - 4π/3)
\(S_{2}\) = 2 \(\cos(C/3 - 4\pi /3) : 1 : 2\) cos(A/3 - 4π/3)
\(S_{3}\) = 2 cos(B/3 - 4π/3) : 2 cos(A/3 - 4π/3) : 1
The triangles M1M2M3, P1P2P3, \(S_{1}\)\(S_{2}\)\(S_{3}\) are discussed in TCCT, page 165-166. The three are pairwise homothetic. See also
1st Morley triangle at MathWorld
2nd Morley triangle at MathWorld
3rd Morley triangle at MathWorld
Unique equilateral triangle inscribed in ABC and homothetic to the 1st Morley triangle, J1J2J3, where
J1 =\(0 : \sin(A/3 - C/3 + \pi /3) : \sin(A/3 - B/3 + \pi /3)\)
J2 =\(\sin(B/3 - C/3 + \pi /3) : 0 : \sin(B/3 - A/3 + \pi /3)\)
J3 =\(\sin(C/3 - B/3 + \pi /3) : \sin(C/3 - A/3 + \pi /3) : 0\)

Circumtangential triangle, inscribed in the circumcircle of ABC and homothetic to the 1st Morley triangle, T1T2T3. For trilinears, of the vertices, see TCCT, page 166, or MathWorld. Trilinears found by M. Stevanovic:

T1 = sin(B/3 - A/3 + π/3) sin(C/3 - A/3 + π/3) : sin(C/3 - B/3) sin(B/3 - A/3 + π/3) : sin(B/3 - C/3) sin(C/3 - A/3 + π/3)
T2 = sin(C/3 - A/3) sin(A/3 - B/3 + π/3) : sin(C/3 - B/3 + π/3) sin(A/3 - B/3 + π/3) : sin(A/3 - C/3) sin(C/3 - B/3 + π/3)
T3 = sin(B/3 - A/3) sin(A/3 - C/3 + π/3) : sin(A/3 - B/3) sin(B/3 - C/3 + π/3) : sin(A/3 - C/3 + π/3) sin(B/3 - C/3 + π/3)

Circumnormal triangle inscribed in the circumcircle of ABC and homothetic to the 1st Morley triangle, N1N2N3. For trilinears, of the vertices, see TCCT, page 166, or MathWorld. Trilinears found by M. Stevanovic:

N1 = - cos(B/3 - A/3 + π/3) cos(C/3 - A/3 + π/3) : cos(B/3 - C/3) cos(B/3 - A/3 + π/3) : cos(B/3 - C/3) cos(C/3 - A/3 + π/3)
N2 = cos(C/3 - A/3) cos(A/3 - B/3 + π/3) : - cos(C/3 - B/3 + π/3) cos(A/3 - B/3 + π/3) : cos(A/3 - A/3) cos(C/3 - B/3 + π/3)
N3 = cos(A/3 - B/3) cos(A/3 - C/3 + π/3) : cos(A/3 - B/3) cos(B/3 - C/3 + π/3) : - cos(A/3 - C/3 + π/3) cos(B/3 - C/3 + π/3)

See also X3602-X3609 and

Jean-Pierre Ehrmann and Bernard Gibert, “A Morley Configuration,” :cite:`Forum Geometricorum <forum-geometricorum>` 1 (2001) 51-58. (Click here to download a PDF of this article.)

Peter Moses notes (April, 14, 2008) that, regarding the reference triangle R and the 6 equilateral triangles (here abbreviated as M, P, S, T, N, J), there are 21 pairs, and each pair except {R,T}, {R,N}, {R, J} are perspective. Perspectors for the remaining 18 pairs are given by this table:

Pair Perspector R, M X357 R, P X1136 R, S X1134 M, P X358 M, S X1135 M, T X3278 M, N X3279 M, J X3273 P, S X1137 P, T X3280 P, N X3281 P, J X3274 S, T X3282 S, N X3283 S, J X3275 T, N X3 T, J X3334 N, J X3335

Centers of the six equilateral triangles are as follows:

Triangle Center M X356 P X3276 S X3277 T X3 N X3 J X3272

X_3232 — 2nd TRISECTED PERIMETER POINT

X32322nd TRISECTED PERIMETER POINT

There exist points A', B', C' on segments BC, CA, AB, respectively, such that B'C + C'B = C'A + A'C = A'B + B'A = (a + b + c)/3, and the lines AA', BB', CC' concur in X3232. Near the beginning of the 21st century, trilinears \(x : y : z\) were found for X3232 in terms of the unique real root of a cubic polynomial related to the cubic polynomial shown at X369, the 1st trisected perimeter point. The proof is given in Sadi Abu-Saymeh, Mawaffaq Hajja, and Hellmuth Stachel, “Another cubic associated with the triangle,” forthcoming in Journal for Geometry and Graphics.