X₂₀₀

X₈-CEVA CONJUGATE OF X₉

X₂₀₀ is the {X₈,X₇₈}-harmonic conjugate of X₁. For a list of harmonic conjugates of X₂₀₀, click Tables at the top of this page.

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

X₂₀₀ (X₈-CEVA CONJUGATE OF X₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 2}, {9, 55}

• Related Centers: X₈ (NAGEL POINT), X₉ (MITTENPUNKT), X₁ (INCENTER), X₂ (CENTROID), X₅₅ (INSIMILICENTER)

Trilinears: \(\cot^{2}(A/2) : :\)

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

Search Value (6, 9, 13): \(4.869827731426\)

▸ Alternate Coordinates4 forms

• Trilinears: \((b + c - a)^{2} : :\)

• Trilinears: \((1 + \cos A)/(1 - \cos A) : :\) (Randy Hutson, 9/23/2011)

• Trilinears: \(1 - \csc^{2}(A/2) : :\)

• Trilinears: squared distance of A to Gergonne line : :

▸ Incident Central Lines18 lines

{1, 2}, {3, 963}, {9, 55}, {33, 281}, {40, 64}, {46, 1004}, {57, 518}, {63, 100}, {69, 269}, {159, 197}, {219, 282}, {220, 728}, {255, 271}, {318, 1089}, {319, 326}, {329, 516}, {341, 1043}, {756, 968}

▸ Functional Relations23 relations

• X₂₀₀ = reflection of X_i in X_j for these (i,j): (1, 997), (57, 1376)

• X₂₀₀ = isogonal conjugate of X₂₆₉

• X₂₀₀ = isotomic conjugate of X₁₀₈₈

• X₂₀₀ = complement of X₃₆₈₄₅

• X₂₀₀ = X₈-Ceva conjugate of X₉

• X₂₀₀ = cevapoint of X₂₂₀ and X₄₈₀

• X₂₀₀ = X₂₂₀-cross conjugate of X₉

• X₂₀₀ = crosspoint of X₈ and X₃₄₆

• X₂₀₀ = crosssum of X_i and X_j for these (i,j): (56, 1407), (57, 1420), (1042, 1427)

• X₂₀₀ = X_i-beth conjugate of X_j for these (i,j): (100, 223), (200, 55), (643, 165)

• X₂₀₀ = {X₁,X₈}-harmonic conjugate of X₄₈₅₃

• X₂₀₀ = {X₂,X₈}-harmonic conjugate of X₄₈₄₇

• X₂₀₀ = homothetic center of anticomplementary triangle and 3rd antipedal triangle of X₁

• X₂₀₀ = homothetic center of ABC and medial triangle of 3rd antipedal triangle of X₁

• X₂₀₀ = Danneels point of X₈

• X₂₀₀ = polar conjugate of X₁₈₄₇

• X₂₀₀ = trilinear square of X₉

• X₂₀₀ = trilinear product of the circumcircle intercepts with the excircles

• X₂₀₀ = X₁₈₉₉-of-excentral-triangle

• X₂₀₀ = complement of polar conjugate of isogonal conjugate of X₂₂₁₅₃

• X₂₀₀ = excentral-to-ABC barycentric image of X₅₇

• X₂₀₀ = mixtilinear-incentral-to-ABC barycentric image of X₁

• X₂₀₀ = mixtilinear-excentral-to-ABC barycentric image of X₁

Notes

Let N_aN_bN_c be the extouch triangle. Let A″ be the trilinear product of the circumcircle intercepts of line N_bN_c, and define B″ and C″ cyclically. The lines N_aA″, N_bB″, N_cC″ concur in X₂₀₀. (Randy Hutson, July 31 2018)

Let A' be the trilinear product of the circumcircle intercepts of the A-excircle. Define B' and C' cyclically. The lines AA', BB', CC' concur in X₂₀₀. (Randy Hutson, July 31 2018)