X₁₈₃₆

ZOSMA TRANSFORM OF X₃₃

X₁₈₃₆ = crosspoint of X₄ and X₇

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

X₁₈₃₆ (ZOSMA TRANSFORM OF X₃₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 30}, {5, 46}

• Related Centers: X₁ (INCENTER), X₃₀ (EULER INFINITY POINT), X₅ (NINE-POINT CENTER), X₄₆ (-CEVA CONJUGATE OF X₁)

Trilinears: \(\cos B + \cos C + 2 \cos B \cos C : :\)

Barycentrics: (y + z) tan A : (z + x) tan B : (x + y) tan C,\(x : y : z\) = X₃₃

Search Value (6, 9, 13): \(-0.622263612144\)

▸ Alternate Coordinates1 form

• Trilinears: (y + z) sec A : (z + x) sec B : (x + y) sec C,\(x : y : z\) = X₃₃

▸ Incident Central Lines30 lines

{1, 30}, {2, 1155}, {3, 1770}, {4, 65}, {5, 46}, {6, 1839}, {7, 354}, {11, 57}, {12, 40}, {19, 1901}, {33, 1892}, {34, 1852}, {55, 226}, {56, 946}, {210, 329}, {221, 225}, {235, 1452}, {278, 1456}, {377, 960}, {381, 1737}, {388, 962}, {517, 1478}, {614, 1086}, {908, 1376}, {942, 1479}, {1040, 1721}, {1158, 1454}, {1399, 1777}, {1470, 1519}, {1826, 1853}

▸ Functional Relations12 relations

• X₁₈₃₆ = reflection of X_i in X_j for these (i,j): (55, 226), (1012, 946)

• X₁₈₃₆ = isogonal conjugate of X₃₇₇₄₁

• X₁₈₃₆ = isotomic conjugate of X₃₄₄₀₉

• X₁₈₃₆ = anticomplement of X₄₆₄₀

• X₁₈₃₆ = crosspoint of X₄ and X₇

• X₁₈₃₆ = crosssum of X_i and X_j for these {i,j}: {3, 55}, {500, 23207}

• X₁₈₃₆ = pole of orthic axis wrt the incircle

• X₁₈₃₆ = crossdifference of every pair of points on line X₉₄₀₄X_34975

• X₁₈₃₆ = {X₄,X₆₅}-harmonic conjugate of X₁₈₃₇

• X₁₈₃₆ = orthologic center of these triangles: 1st Johnson-Yff to 3rd extouch

• X₁₈₃₆ = X₄₀-of-1st-Johnson-Yff-triangle

• X₁₈₃₆ = outer-Johnson-to-ABC similarity image of X₄₀

Notes

Let A'B'C' be the intangents triangle. Let A″ be the reflection of A' in BC, and define B″ and C″ cyclically. A″B″C″ is perspective to ABC at X₁₈₃₆. (Randy Hutson, June 7, 2019)