X₁₈₇

INVERSE-IN-CIRCUMCIRCLE OF X₆ (SCHOUTE CENTER)

Let L denote the line having trilinears of X₁₈₇ as coefficients. Then L is the line passing through X₂ perpendicular to the Euler line.

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

X₁₈₇ (INVERSE-IN-CIRCUMCIRCLE OF X₆ (SCHOUTE CENTER)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Circle: Circumcircle (circumcircle) — Passes through vertices A, B, C; centered at X₃ CIRCUMCENTER

• Central Lines: {3, 6}

• Related Centers: X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT)

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

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

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

Search Value (6, 9, 13): \(-1.840883737592\)

▸ Alternate Coordinates4 forms

• Trilinears: \(\sin A - 3 \cos A \tan \omega : :\)

• Trilinears: \(2 \sin(A - 2\omega ) - \sin(A + 2\omega ) + \sin A : :\)

• Trilinears: \(\sin A + \sin A \cos 2\omega - 3 \cos A \sin 2\omega : :\)

• Trilinears: \(\cos(A + \omega ) \sin 2\omega - e^{2} \sin(A - \omega ) : :\)

▸ Linear Combinations1 combination

• X₁₈₇ = X₁₅ + X₁₆

X₁₈₇ lies on the Darboux quintic.

▸ Incident Central Lines18 lines

{2, 316}, {3, 6}, {23, 111}, {30, 115}, {35, 172}, {36, 1015}, {74, 248}, {99, 385}, {110, 352}, {112, 186}, {183, 1003}, {237, 351}, {249, 323}, {325, 620}, {353, 3117}, {395, 531}, {396, 530}, {729, 805}

▸ Functional Relations52 relations

• X₁₈₇ = midpoint of X_i and X_j for these (i,j): (15, 16), (99, 385)

• X₁₈₇ = reflection of X_i in X_j for these (i,j): (115, 230), (316, 625), (325, 620), (5107, 6)

• X₁₈₇ = isogonal conjugate of X₆₇₁

• X₁₈₇ = isotomic conjugate of X₁₈₀₂₃

• X₁₈₇ = inverse-in-circumcircle of X₆

• X₁₈₇ = inverse-in-Brocard-circle of X₅₇₄

• X₁₈₇ = inverse-in-van-Lamoen-circle-of-X₂

• X₁₈₇ = radical trace of the circumcircle and Brocard circle

• X₁₈₇ = complement of X₃₁₆

• X₁₈₇ = anticomplement of X₆₂₅

• X₁₈₇ = X_i-Ceva conjugate of X_j for these (i,j): (2, 6593), (111, 6)

• X₁₈₇ = crosspoint of X_i and X_j for these (i,j): (2, 67), (6, 111), (468, 524)

• X₁₈₇ = crosssum of X_i and X_j for these (i,j): (2, 524), (6, 23), (111, 895), (115, 690)

• X₁₈₇ = crossdifference of every pair of points on line X₂X₅₂₃

• X₁₈₇ = X₅₅-beth conjugate of X₁₈₇

• X₁₈₇ = inverse-in-Moses-radical-circle of X₁₄₉₅

• X₁₈₇ = radical trace of Moses radical circle and Parry circle

• X₁₈₇ = radical trace of Lucas radical circle and Lucas(-1) radical circle

• X₁₈₇ = radical trace of Lucas inner and Lucas(-1) inner circle

• X₁₈₇ = radical trace of circles {P(1),U(2),U(39)}} and {U(1),P(2),P(39)}}

• X₁₈₇ = intersection of Brocard axis and Lemoine axis

• X₁₈₇ = intersection of Brocard axis (or Lemoine axis) and non-transverse axis of hyperbola {A, B, C,PU(2)}}

• X₁₈₇ = intersection of Brocard axis (or Lemoine axis) and tangent at X₆₉₁ to hyperbola {A, B, C,PU(2)}}

• X₁₈₇ = midpoint of PU(2)

• X₁₈₇ = bicentric sum of PU(2)

• X₁₈₇ = perspector of ABC and the reflection of the circumsymmedial triangle in the Brocard axis

• X₁₈₇ = perspector of ABC and the reflection of the circumsymmedial triangle in the Lemoine axis

• X₁₈₇ = intersection of tangents to hyperbola {A, B, C,X₂,X₆} at X₆ and X₁₁₁

• X₁₈₇ = inverse-in-Parry-circle of X₂₅₀₂

• X₁₈₇ = X₁₈₇-of-2nd-Brocard-triangle

• X₁₈₇ = X₁₈₇-of-circumsymmedial-triangle

• X₁₈₇ = polar conjugate of isogonal conjugate of X₂₃₂₀₀

• X₁₈₇ = polar conjugate of isotomic conjugate of X₃₂₉₂

• X₁₈₇ = X₉₂-isoconjugate of X₈₉₅

• X₁₈₇ = X₁₅₇₇-isoconjugate of X₆₉₁

• X₁₈₇ = {X₁₆₈₇,X₁₆₈₈}-harmonic conjugate of X₂₀₈₀

• X₁₈₇ = trilinear pole of PU(107)

• X₁₈₇ = inverse-in-Parry-isodynamic-circle of X₃₅₁; see X₂

• X₁₈₇ = radical trace of 3rd and 4th Lozada circles

• X₁₈₇ = radical trace of 6th and 7th Lozada circles

• X₁₈₇ = radical trace of 8th and 9th Lozada circles

• X₁₈₇ = radical trace of 10th and 11th Lozada circles

• X₁₈₇ = radical trace of circumcircles of outer and inner Grebe triangles

• X₁₈₇ = X₁₁₅-of-4th-anti-Brocard-triangle

• X₁₈₇ = X₁₈₇-of-X₃PU(1)

• X₁₈₇ = Thomson-isogonal conjugate of X₆₀₅₄

• X₁₈₇ = Cundy-Parry Phi transform of X₅₇₆

• X₁₈₇ = Cundy-Parry Psi transform of X₇₆₀₇

• X₁₈₇ = X₁₃-antipedal-to-X₁₆-pedal similarity image of X₁₃

• X₁₈₇ = X₁₄-antipedal-to-X₁₅-pedal similarity image of X₁₄

• X₁₈₇ = homothetic center of Trinh triangle and mid-triangle of 1st and 2nd Kenmotu diagonals triangles

• X₁₈₇ = QA-P₄ (Isogonal Center of quadrangle ABCX₂; see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/25-qa-p4.html)

Notes

Let A'B'C' be the 1st Brocard triangle. Let A″B″C″ be the 2nd Brocard triangle. Let A* = Λ((A',A″), and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X₁₈₇. (Randy Hutson, December 26, 2015)

Let (O_A) be the circumcircle of BCX₂. Let P_A be the perspector of (O_A). Let L_A be the polar of P_A wrt (O_A). Define L_B and L_C cyclically. Let A' = L_B∩L_C, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₁₈₇. (Randy Hutson, June 7, 2019)

X₁₈₇ is the {X₃,X₆}-harmonic conjugate of X₅₇₄.