X₄₈₂

2nd EPPSTEIN POINT

Let S' be the outer Soddy circle and \(S_{A}\), \(S_{B}\), \(S_{C}\) the Soddy circles tangent to S. Let J_A = S'∩\(S_{A}\), E_A = \(S_{B}\)∩\(S_{C}\), and determine J_B, J_C, E_B, E_C cyclically. Then X₄₈₂ is the point of concurrence of lines J_A-to-E_A, J_B-to-E_B, J_C-to-E_C.

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

X₄₈₂ (2nd EPPSTEIN POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 7}, {226, 486}

• Related Centers: X₇₉ (ISOGONAL CONJUGATE OF X₃₅), X₄₈₁ (1st EPPSTEIN POINT), X₁ (INCENTER), X₇ (GERGONNE POINT), X₂₂₆ (-CEVA CONJUGATE OF X₆₅), X₄₈₆ (INNER VECTEN POINT)

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

Barycentrics: \(a + (a + b + c) \tan(A/2) : :\)

Search Value (6, 9, 13): \(1.036212579546\)

▸ Alternate Coordinates1 form

• Trilinears: \(1 + 4(\operatorname{area})/[a(b + c - a)] : :\) (E. Brisse, 3/20/01)

▸ Linear Combinations3 combinations

• X₄₈₂ = \(s\) X₁ + \((r + 4R)\) X₇

• X₄₈₂ = A / \(R_{a}\) + B / \(R_{b}\) + C / \(R_{c}\) + X₁ / \(r\), where \(R_{a}\), \(R_{b}\), \(R_{c}\) = radii of Soddy circles

• X₄₈₂ = A / \(R_{a}\) + B / \(R_{b}\) + C / \(R_{c}\) + X₁₇₆ / \(R_{s}\), where \(R_{a}\), \(R_{b}\), \(R_{c}\) = radii of Soddy circles, and \(R_{s}\) = radius of inner Soddy circle

▸ Incident Central Lines2 lines

{1, 7}, {226, 486}

▸ Functional Relations7 relations

• X₄₈₂ = X₇₉-Ceva conjugate of X₄₈₁

• X₄₈₂ = trilinear product of centers of Elkies companion incircles

• X₄₈₂ = Kosnita(X₁₇₆,X₇) point

• X₄₈₂ = Kosnita(X₁₇₆,X₁₇₆) point

• X₄₈₂ = {X₇,X₁₇₅}-harmonic conjugate of X₁₃₇₄

• X₄₈₂ = 2nd-inner-Soddy-isogonal conjugate of X₃₂₀₅₇

• X₄₈₂ = perspector of ABC and the intangents triangle of the three Elkies companion incircles; see X₁₇₆

Notes

David Eppstein, “Tangent spheres and triangle centers,” American Mathematical Monthly, 108 (2001) 63-66.

Let S be the inner Soddy center, X₁₇₆. X₄₈₂ is the point of concurrence of the Soddy lines of BSC, CSA, and ASB. (Randy Hutson, September 14, 2016)

Let I_a, I_b, I_c be the centers of the Elkies companion incircles. Let A' be the trilinear product I_b*I_c, and define B' and C’cyclically. The lines AA', BB', CC' concur in X₄₈₂. The lines I_aA', I_bB', I_cC' concur in X₁₇₆. (Randy Hutson, September 14, 2016)

Let L_a be the line tangent to the inner Soddy circle at the touchpoint with the A-Soddy circle. Define L_b and L_c cyclically. Let A' = L_b∩L_c and define B' and C' cyclically. Triangle A'B'C', here introduced as the inner Soddy tangential triangle, is perspective to ABC, and the perspector is X₄₈₂. (Randy Hutson, September 14, 2016)