X₉₄₂

INCIRCLE-INVERSE OF X₃₆

X₉₄₂ is the incircle-inverse of X₃₆.

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

X₉₄₂ (INCIRCLE-INVERSE OF X₃₆) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 65}

• Related Centers: X₁ (INCENTER), X₆₅ (ORTHOCENTER OF THE INTOUCH TRIANGLE)

Trilinears: \(2abc + (b + c)(a - b + c)(a + b -c) : :\)

Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = af(a,b,c)\)

Search Value (6, 9, 13): \(1.110017411481\)

▸ Alternate Coordinates1 form

• Trilinears: \(1 + \cos B + \cos C : :\)

▸ Incident Central Lines27 lines

{1, 3}, {2, 72}, {4, 7}, {5, 226}, {6, 169}, {8, 443}, {10, 141}, {11, 113}, {28, 60}, {30, 553}, {34, 222}, {37, 579}, {42, 1066}, {58, 1104}, {63, 405}, {78, 474}, {212, 582}, {238, 1046}, {277, 1002}, {279, 955}, {284, 501}, {355, 388}, {496, 946}, {750, 976}, {758, 960}, {962, 1058}, {1042, 1064}

▸ Functional Relations21 relations

• X₉₄₂ = midpoint of X₁ and X₆₅

• X₉₄₂ = reflection of X₉₆₀ in X₁₁₂₅

• X₉₄₂ = isogonal conjugate of X₉₄₃

• X₉₄₂ = complement of X₇₂

• X₉₄₂ = anticomplement of X₅₀₄₄

• X₉₄₂ = complementary conjugate of X₂₁₅₃₀

• X₉₄₂ = incircle-inverse of X₃₆

• X₉₄₂ = Conway-circle-inverse of X₃₈₄₈₃

• X₉₄₂ = X₁-Ceva conjugate of X₅₀₀

• X₉₄₂ = crosspoint of X_i and X_j for these (i,j): (1, 79), (2, 286), (7, 81)

• X₉₄₂ = crosssum of X_i and X_j for these (i,j): (1, 35), (6, 228), (37, 55)

• X₉₄₂ = X₅-of-the-intouch-triangle

• X₉₄₂ = X₁₁₁₂ of Fuhrmann triangle

• X₉₄₂ = {X₁,X₄₀}-harmonic conjugate of X₃₂₉₅

• X₉₄₂ = X₃-of-inverse-in-incircle-triangle

• X₉₄₂ = X₄-of-incircle-circles-triangle

• X₉₄₂ = X₆₇₅₆-of-excentral-triangle

• X₉₄₂ = excentral-to-intouch similarity image of X₃

• X₉₄₂ = Cundy-Parry Phi transform of X₅₅

• X₉₄₂ = Cundy-Parry Psi transform of X₇

• X₉₄₂ = QA-P23 (Inscribed Square Axes Crosspoint) of quadrangle ABCX₁ (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/51-qa-p23.html)

Notes

Let I_aI_bI_c be the incentral triangle of triangle ABC. Let L_A be the line of reflection of line BC in line I_bI_c, and define L_B and L_C cyclically. Let A″ = L_B∩L_C, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₉₄₂. (Randy Hutson, 9/23/2011)

Let I be the incenter of ABC. Let N_A be the nine-point center of the triangle IBC, and define N_B and N_C cyclically. Let A_A be the reflection of N_A in the line AI, Let A_B be the reflection of N_A in the line BI, let Let A_C be the reflection of N_A in the line CI, and define B_A, B_B, and B_C cyclically, and define C_A, C_B, and C_C cyclically. Let O_A be the nine-point center of the triangle A_AA_BA_C, let O_B be the nine-point center of the triangle B_AB_BB_C, and let O_C be the nine-point center of the triangle C_AC_BC_C. The circles O_A, O_B, O_C concur in X₉₄₂. (Antreas Hatzipolakis and César Lozada, January 28, 2015, Hyacinthos 23074)

X₉₄₂ is the center of the conic which is the locus of poles, wrt the incircle, of tangents to the circumcircle. This conic has foci at X₁ and X₆₅. (Randy Hutson, July 20, 2016)

Let T_aT_bT_c be the intouch triangle. Let A″ be the orthocenter of AT_bT_c, and define B″ and C″ cyclically. The triangle A″B″C″ is homothetic to T_aT_bT_c, and the center of homothety is X₉₄₂. (Randy Hutson, July 20, 2016)

Let H_aH_bH_c be the orthic triangle. Let A″ be the incenter of AH_bH_c, and define B″ and C″ cyclically. The triangle A″B″C″ is homothetic to the intouch triangle, and the center of homothety is X₉₄₂. (Randy Hutson, July 20, 2016)

Let A' be the homothetic center of the orthic triangles of the intouch and A-extouch triangles, and define B' and C' cyclically. The triangle A'B'C' is perspective to the half-altitude triangle at X₉₄₂. In fact, A'B'C' is the cevian triangle of X₉₄₂, wrt the half-altitude triangle. (Randy Hutson, July 20, 2016)

A construction of X₉₄₂ is given at 24080. (Antreas Hatzipolakis, August 29, 2016)