X₈₀

REFLECTION OF INCENTER IN FEUERBACH POINT

X₈₀ is the reflection of incenter in Feuerbach point.

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

X₈₀ (REFLECTION OF INCENTER IN FEUERBACH POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 11}

• Related Centers: X₁ (INCENTER), X₁₁ (FEUERBACH POINT)

Trilinears: \(1/(1 - 2 \cos A) : 1/(1 - 2 \cos B) : 1/(1 - 2 \cos C)\)

Barycentrics: \(1/(b^{2} + c^{2} - a^{2} - bc) : :\)

Search Value (6, 9, 13): \(3.595961733033\)

▸ Alternate Coordinates3 forms

• Trilinears: \(bc/(b^{2} + c^{2} - a^{2} - bc) : :\)

• Trilinears: \(\cos(A/2) \sec(3A/2) : :\)

• Barycentrics: \(1/(bc - 2 S_{A}) : :\)

▸ Linear Combinations1 combination

• X₈₀ = \((2r + R)\) X₁ - \(6r\) X₂ + \(2r\) X₃ (Peter Moses, April 2, 2013)

▸ Incident Central Lines19 lines

{1, 5}, {2, 214}, {7, 150}, {8, 149}, {9, 528}, {10, 21}, {30, 484}, {33, 1061}, {34, 1063}, {36, 104}, {40, 90}, {46, 84}, {65, 79}, {313, 314}, {497, 1000}, {499, 944}, {516, 655}, {519, 908}, {943, 950}

▸ Functional Relations26 relations

• X₈₀ = midpoint of X₈ and X₁₄₉

• X₈₀ = reflection of X_i in X_j for these (i,j): (1, 11), (100, 10), (1317, 1387)

• X₈₀ = isogonal conjugate of X₃₆

• X₈₀ = isotomic conjugate of X₃₂₀

• X₈₀ = circumcircle-inverse of X₁₀₂₆₀

• X₈₀ = incircle-inverse of X₁₃₈₇

• X₈₀ = Fuhrmann-circle-inverse of X₁

• X₈₀ = complement of X₆₂₂₄

• X₈₀ = anticomplement of X₂₁₄

• X₈₀ = cevapoint of X₁₀ and X₅₁₉

• X₈₀ = X_i-cross conjugate of X_j for these (i,j): (44, 2), (517, 1)

• X₈₀ = X₈-beth conjugate of X₁₀₀

• X₈₀ = antigonal image of X₁

• X₈₀ = syngonal conjugate of X₁₀

• X₈₀ = X₁₈₆-of-Fuhrmann triangle

• X₈₀ = orthology center of ABC and Fuhrmann triangle

• X₈₀ = reflection of any vertex of ABC in the corresponding side of the Fuhrmann triangle

• X₈₀ = perspector of ABC and reflection of Fuhrmann triangle in X₁₁

• X₈₀ = trilinear pole of line X₃₇X₆₅₀

• X₈₀ = inverse-in-circumconic-centered-at-X₁-of-X₁₈₀₇

• X₈₀ = perspector of ABC and extraversion triangle of X₇₉

• X₈₀ = X₁₉₈₆-of-excentral triangle

• X₈₀ = perspector of ABC and mid-triangle of 1st and 2nd extouch triangles

• X₈₀ = inner-Garcia-to-outer-Garcia similarity image of X₁

• X₈₀ = X₁₀₀-of-outer-Garcia-triangle

• X₈₀ = Conway-circle-inverse of X₃₅₆₃₈

Notes

Let A' be the reflection in BC of the A-vertex of the excentral triangle, and define B' and C' cyclically. The circumcircles of A'BC, B'CA, and C'AB concur in X₈₀. Also, the lines AA', BB', CC' concur in X₈₀. (Randy Hutson, December 10, 2016)

Let A'B'C' be the Fuhrmann triangle. Let L_a be the line through A parallel to B'C', and define L_b and L_c cyclically. Let A″ = L_b∩L_c, B″ = L_c∩L_a, C″ = L_a∩L_b. The lines A'A″, B'B″, C'C″ concur in X₈₀. (Randy Hutson, December 10, 2016)

Let H_aH_bH_c be the orthic triangle. Let L_a be the antiorthic axis of AH_bH_c, and define L_b and L_c cyclically. Let A″ = L_b∩L_c. B″ = L_c∩L_a, C″ = L_a∩L_b. The triangle A″B″C″ is inversely similar to ABC, with similitude center X₉. The incenter of triangle A″B″C″ is X₈₀. Also, the lines AA″, BB″, CC″ concur in X₈₀.(Randy Hutson, December 10, 2016)

Let J_aJ_bJ_c be the excentral triangle. Let A″ be the isogonal conjugate, wrt J_aBC, of A. Define B″, C″ cyclically. (A″ is also the reflection of J_a in BC, and cyclically for B″ and C″). The lines AA″, BB″, CC″ concur in X₈₀. (Randy Hutson, January 29, 2018)

Let J_aJ_bJ_c be the excentral triangle. Let O_a be the J_a-Johnson circle of triangle J_aBC, and define O_b and O_c cyclically. X₈₀ is the radical center of O_a, O_b, O_c. (Randy Hutson, June 27, 2018)