X17

1st NAPOLEON POINT

Let Napa, Napb, Napc be the centers of the equilateral triangles in the construction of X13. The lines ANapa, BNapb, CNapc concur in X17.
< A B C >: Reference Triangle △ABC < B C A' >: Equilateral Triangle on BC < C A B' >: Equilateral Triangle on CA < A B C' >: Equilateral Triangle on AB < Nap_a Nap_b Nap_c >: Outer Napoleon Triangle △Nap_a Nap_b Nap_c [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) [ A Nap_a ]: Outer Napoleon Cevian A–Nap_a [ B Nap_b ]: Outer Napoleon Cevian B–Nap_b [ C Nap_c ]: Outer Napoleon Cevian C–Nap_c { 3, 13 }: Central Line L(3, 13) { 4, 15 }: Central Line L(4, 15) { 5, 14 }: Central Line L(5, 14) { 6, 18 }: Central Line L(6, 18) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) A' = Auxiliary Apex A' (1.3256, 0.6357) B' = Auxiliary Apex B' (-0.2134, 0.8302) C' = Auxiliary Apex C' (0.5000, -0.8660) Nap_a = Napoleon Center Nap_a (0.9793, 0.4119) Nap_b = Napoleon Center Nap_b (0.1330, 0.4767) Nap_c = Napoleon Center Nap_c (0.5000, -0.2887) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_13 = 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT) (0.5879, 0.2819) X_4 = ORTHOCENTER (0.6122, 0.3958) X_15 = 1st ISODYNAMIC POINT (0.5519, 0.1878) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_14 = 2nd ISOGONIC CENTER (0.9134, -0.1143) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_18 = 2nd NAPOLEON POINT (0.4434, 0.1317) X_17 = 1st NAPOLEON POINT (0.5666, 0.2383) A B C A' B' C' Napa Napb Napc X3 X13 X4 X15 X5 X14 X6 X18 X17

X17 (1st NAPOLEON POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

  • Equilateral Side Triangles: △BCA', △CAB', △ABC' erected on sidelines with apexes A', B', C'

  • Derived Triangle: Outer Napoleon Triangle △NapaNapbNapc — vertices Napa, Napb, Napc (Equilateral triangle formed by centers of outward side equilateral triangles)

  • Concurrence Lines: A–Napa, B–Napb, C–Napc concurring at X17

Trilinears: \(\csc(A + \pi /6) : \csc(B + \pi /6) : \csc(C + \pi /6)\)
Barycentrics: \(a \csc(A + \pi /6) : b \csc(B + \pi /6) : c \csc(C + \pi /6)\)
Tripolars: \((a^{2}-b^{2}-c^{2}-2 \sqrt{3} S) \sqrt{(a^{2}-3 b^{2}-3 c^{2}-2 \sqrt{3} S)} : :\)
Search Value (6, 9, 13): \(1.386654550070\)
▸ Alternate Coordinates1 form

  • Trilinears: \(\sec(A - \pi /3) : \sec(B - \pi /3) : \sec(C - \pi /3)\)

X17 lies on the Napoleon cubic.

Related Collection 1 Topic Guides

T_13: The Isogonic & Napoleon Central Lines • T_24: Equilateral, Isogonic & Napoleon Triangles

▸ Incident Central Lines13 lines

{2, 62}, {3, 13}, {4, 15}, {5, 14}, {6, 18}, {12, 203}, {16, 140}, {76, 303}, {83, 624}, {202, 499}, {275, 471}, {299, 635}, {623, 633}

▸ Functional Relations15 relations

Notes

John Rigby, “Napoleon revisited,” Journal of Geometry,33 (1988) 126-146.

X17 is the {X231,X1209}-harmonic conjugate of X18.