X₁₈₅

NAGEL POINT OF THE ORTHIC TRIANGLE

Alexei Myakishev has noted that X₁₈₅ is the Nagel point of the orthic triangle only is ABC is an acute triangle.

5, 6, 7 3, 4, 5 Kepler Kimberling

X₁₈₅ (NAGEL POINT OF THE ORTHIC TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 49}, {4, 51}

• Related Centers: X₁₁₀₅ (CEVAPOINT OF X₃ AND X₄), X₃ (CIRCUMCENTER), X₄₉ (CENTER OF SINE-TRIPLE-ANGLE CIRCLE), X₄ (ORTHOCENTER), X₅₁ (CENTROID OF ORTHIC TRIANGLE)

Trilinears: \((\cos A)[1 - \cos A \cos(B - C)] : :\)

Barycentrics: \((\sin 2A)[1 - \cos A \cos(B - C)] : (\sin 2B)[1 - \cos B \cos(C - A)] : (\sin 2C)[1 - \cos C \cos(A - B)]\)

Search Value (6, 9, 13): \(11.715115569976\)

▸ Alternate Coordinates2 forms

• Trilinears: \((\cos A)(\cos^{2} B + \cos^{2} C) : :\)

• Trilinears: \(a(b^{2} + c^{2} - a^{2})[2a^{2}(b^{2} - c^{2})^{2} - a^{4}(b^{2} + c^{2}) - (b^{2} - c^{2})^{2}(b^{2} + c^{2})] : :\)

▸ Incident Central Lines16 lines

{1, 296}, {3, 49}, {4, 51}, {5, 113}, {6, 64}, {20, 193}, {25, 1498}, {30, 52}, {39, 217}, {54, 74}, {72, 916}, {287, 384}, {378, 578}, {382, 568}, {411, 970}, {648, 1105}

▸ Functional Relations21 relations

• X₁₈₅ = reflection of X_i in X_j for these (i,j): (4, 389), (125, 974)

• X₁₈₅ = isogonal conjugate of X₁₁₀₅

• X₁₈₅ = X_i-Ceva conjugate of X_j for these (i,j): (3, 417), (4, 235)

• X₁₈₅ = crosspoint of X₃ and X₄

• X₁₈₅ = crosssum of X_i and X_j for these (i,j): (3, 4), (25, 1249)

• X₁₈₅ = anticomplement of X₅₉₀₇

• X₁₈₅ = bicentric sum of PU(17)

• X₁₈₅ = PU(17)-harmonic conjugate of X₆₄₇

• X₁₈₅ = orthology center of orthic and half-altitude triangles

• X₁₈₅ = half-altitude isogonal conjugate of X₄

• X₁₈₅ = orthic-isogonal conjugate of X₂₃₅

• X₁₈₅ = orthic-isotomic conjugate of X₁₈₄₃

• X₁₈₅ = X₂₀-of-X₄-Brocard-triangle

• X₁₈₅ = anticomplement of X₄ wrt orthic triangle

• X₁₈₅ = X₄-of-tangential-triangle-of-Jerabek-hyperbola

• X₁₈₅ = eigencenter of cevian triangle of X₆₄₈

• X₁₈₅ = eigencenter of anticevian triangle of X₆₄₇

• X₁₈₅ = trilinear product of vertices of 2nd Hyacinth triangle

• X₁₈₅ = X₁₀-of-circumorthic-triangle if ABC is acute

• X₁₈₅ = excentral-to-ABC functional image of X₈

• X₁₈₅ = polar-circle-inverse of X₃₄₁₇₀

Notes

Let H_a be the foot of the A-altitude. Let B_a and C_a be the feet of perpendiculars from H_a to CA and AB, respectively. Let G_a be the centroid of H_aB_aC_a. Define G_b and G_c cyclically. The lines H_aG_a, H_bG_b, H_cG_c concur in X₁₈₅. (Randy Hutson, December 26, 2015)

Let H_a, H_b, H_c be the orthocenters of the A-, B-, and C-altimedial triangles. X₁₈₅ is the orthocenter of H_aH_bH_c. (Randy Hutson, March 25, 2016)

Let P be a point on the circumcircle. Let P_a be the orthogonal projection of P on the A-altitude, and define P_b, P_c cyclically. The locus of the orthocenter of P_aP_bP_c as P varies is an ellipse centered at X₁₈₅. See also X₉₇₃₀. (Randy Hutson, March 25, 2016)

Let A'B'C' be the midheight triangle. Let A_B, A_C be the orthogonal projections of A' on CA, AB, resp. Define B_C, B_A, C_A, C_B cyclically. Let A″ = C_AA_C∩A_BB_A, and define B″ and C″ cyclically. Triangle A″B″C″ is homothetic to ABC at X₆ and perspective to the orthic triangle at X₁₈₅. (Randy Hutson, March 29, 2020)