X₂₃₄₅

X₂-ISOCONJUGATE OF X₁₄₇₂

X₂₃₄₅ = isogonal conjugate of X₂₂₂₁

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

X₂₃₄₅ (X₂-ISOCONJUGATE OF X₁₄₇₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Medial Triangle △M_aM_bM_c — vertices M_a, M_b, M_c (Formed by the midpoints of sides BC, CA, AB)

• Central Lines: {2, 3672}

• Related Centers: X₂ (CENTROID), X₃₆₇₂ (com(INTOUCH TRIANGLE))

Trilinears: \(a/f(a,b,c) : b/f(b,c,a) : c/f(c,a,b)\), where f(a,b,c) is as in X₁₄₇₂

Barycentrics: \(a^{2}/f(a,b,c) : b^{2}/f(b,c,a): c^{2}/f(c,a,b)\)

Search Value (6, 9, 13): \(3.024937548881\)

▸ Alternate Coordinates2 forms

• Barycentrics: \(b c + S_{\omega} : :\)

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

▸ Incident Central Lines28 lines

{1, 2321}, {2, 37}, {4, 9}, {6, 8}, {7, 141}, {44, 391}, {45, 1213}, {48, 2329}, {69, 894}, {145, 1100}, {193, 319}, {198, 1376}, {219, 1065}, {220, 965}, {318, 393}, {329, 1211}, {333, 1778}, {388, 2285}, {404, 2178}, {498, 1733}, {519, 1449}, {572, 944}, {936, 2324}, {948, 1441}, {992, 2176}, {1010, 2303}, {1400, 1788}, {1698, 1738}

▸ Functional Relations8 relations

• X₂₃₄₅ = isogonal conjugate of X₂₂₂₁

• X₂₃₄₅ = complement of X₃₆₇₂

• X₂₃₄₅ = anticomplement of X₄₆₅₇

• X₂₃₄₅ = X₁₀₁₀-Ceva conjugate of X₆₁₂

• X₂₃₄₅ = X₆₁₂-cross conjugate of X₃₈₈

• X₂₃₄₅ = crosspoint of X₂ and X₁₂₁₉

• X₂₃₄₅ = crosssum of X_i and X_j for these (i,j): (6, 1191), (1245, 2281)

• X₂₃₄₅ = {X₂,X₇₅}-harmonic conjugate of X₄₀₀₀