X1656

INTERSECTION OF EULER LINE AND LINEX(17)X18

Let A' be the reflection of X3 in A, and define B' and C' cyclically. The triangle A'B'C' is homothetic to the medial triangle, and the center of homothety is X1656.
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 5, 632 }: Euler Line { 2, 3 }: Euler Line { 6, 17 }: Central Line L(6, 17) { 11, 498 }: Central Line L(11, 498) { 12, 499 }: Central Line L(12, 499) { 125, 399 }: Central Line L(125, 399) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_632 = 9/10*OG (0.5337, 0.1902) X_2 = CENTROID (0.5374, 0.2000) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_17 = 1st NAPOLEON POINT (0.5666, 0.2383) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_498 = YFF CONCURRENT CONGRUENT CIRCLES POINT (0.5518, 0.2140) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5641, 0.2407) X_499 = {X(1),X(2)}-HARMONIC CONJUGATE OF X(498) (0.4458, 0.1102) X_125 = CENTER OF JERABEK HYPERBOLA (0.6803, 0.0260) X_399 = PARRY REFLECTION POINT (0.0032, 0.9936) X_1656 = INTERSECTION OF EULER LINE AND LINEX(17)X(18) (0.5449, 0.2195) A B C X5 X632 X2 X3 X6 X17 X11 X498 X12 X499 X125 X399 X1656 (a, b, c) = (5, 6, 7) | AB = 1

X1656 (INTERSECTION OF EULER LINE AND LINEX(17)X18) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(3 \cos A + 4 \cos B \cos C : :\)
Barycentrics: \(3 + \cot B \cot C : :\)
Search Value (6, 9, 13): \(1.798769972147\)
▸ Alternate Coordinates2 forms

  • Barycentrics: \(3 S^{2} + S_{B} S_{C} : :\)

  • Barycentrics: \(a^{4} - 3a^{2}(b^{2} + c^{2}) + 2(b^{2} - c^{2})^{2} : :\)

▸ Linear Combinations1 combination
▸ Incident Central Lines18 lines

{2, 3}, {6, 17}, {10, 1482}, {11, 498}, {12, 499}, {49, 569}, {51, 1216}, {125, 399}, {141, 1351}, {302, 634}, {303, 633}, {355, 1125}, {373, 568}, {485, 615}, {486, 590}, {517, 1698}, {567, 1147}, {576, 599}

▸ Functional Relations17 relations

  • X1656 = midpoint of X5 and X632

  • X1656 = reflection of Xi in Xj for these (i,j): (3, 631), (631, 632)

  • X1656 = complement of X631

  • X1656 = anticomplement of X632

  • X1656 = circumcircle-inverse of X37936

  • X1656 = orthocentroidal-circle-inverse of X140

  • X1656 = {X3,X5}-harmonic conjugate of X381

  • X1656 = {X17,X18}-harmonic conjugate of X6

  • X1656 = {X1506,X7746}-harmonic conjugate of X6

  • X1656 = homothetic center of 2nd Euler triangle and mid-triangle of orthic and circumorthic triangles

  • X1656 = homothetic center of submedial triangle and mid-triangle of orthic and circumorthic triangles

  • X1656 = X5-of-cross-triangle-of-Euler-and-anti-Euler-triangles

  • X1656 = X1385-of-cross-triangle-of Aquila-and-anti-Aquila-triangles

  • X1656 = homothetic center of X5-altimedial and X3-anti-altimedial triangles

  • X1656 = homothetic center of X140-altimedial and X140-anti-altimedial triangles

  • X1656 = endo-homothetic center of Ehrmann mid-triangle and X3-ABC reflections triangle; the homothetic center is X3843

  • X1656 = radical center of de Longchamps circles of ABC and 1st and 2nd Ehrmann circumscribing triangles

Notes

As a point on the Euler line, X1656 has Shinagawa coefficients \((3,1)\).