X₄₀₃

X₃₆ OF THE ORTHIC TRIANGLE

X₄₀₃ = midpoint of X₄ and X₁₈₆

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

X₄₀₃ (X₃₆ OF THE ORTHIC TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Orthic Triangle △H_aH_bH_c — vertices H_a, H_b, H_c (Pedal triangle of Orthocenter X₄ (altitude feet))

• Concurrence Lines: H_a–X₄₀₃, H_b–X₄₀₃, H_c–X₄₀₃ concurring at X₄₀₃

• Central Lines: {2, 3} (Euler line)

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER)

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

Barycentrics: \((\tan A)(1 + \cos 2B + \cos 2C) : :\)

Search Value (6, 9, 13): \(-1.086449827044\)

▸ Incident Central Lines4 lines

{2, 3}, {112, 230}, {115, 232}, {847, 1093}

▸ Functional Relations24 relations

• X₄₀₃ = midpoint of X₄ and X₁₈₆

• X₄₀₃ = reflection of X₁₈₆ in X₄₆₈

• X₄₀₃ = isogonal conjugate of X₅₅₀₄

• X₄₀₃ = complement of X₂₀₇₁

• X₄₀₃ = anticomplement of X₁₀₂₅₇

• X₄₀₃ = inverse-in-circumcircle of X₂₄

• X₄₀₃ = inverse-in-nine-point-circle of X₄

• X₄₀₃ = inverse-in-orthocentroidal-circle of X₃₇₈

• X₄₀₃ = anticomplementary-circle-inverse of X₃₄₉₃₈

• X₄₀₃ = X₁₁₃-cross conjugate of X₄

• X₄₀₃ = crossdifference of every pair of points on line X₅₇₇X₆₄₇

• X₄₀₃ = X₃₆-of-orthic-triangle if and only if ABC is acute

• X₄₀₃ = X₁₈₆-of Euler-triangle

• X₄₀₃ = {X₄,X₅}-harmonic conjugate of X₁₅₉₄

• X₄₀₃ = inverse-in-polar-circle of X₃

• X₄₀₃ = inverse-in-{circumcircle, nine-point circle}}-inverter of X₂₅

• X₄₀₃ = reflection of X₁₈₆ in the orthic axis

• X₄₀₃ = pole wrt polar circle of trilinear polar of X₂₉₈₆

• X₄₀₃ = X₄₈-isoconjugate (polar conjugate) of X₂₉₈₆

• X₄₀₃ = perspector of the orthic triangle and the reflection of the Euler triangle in the Euler line

• X₄₀₃ = radical trace of polar and second Droz-Farny circles

• X₄₀₃ = inverse-in-second-Droz-Farny-circle of X₄

• X₄₀₃ = excentral-to-ABC functional image of X₃₆

• X₄₀₃ = Euler line intercept, other than X₄₀₃, of circle {X₄₀₃,X₈₅₈,PU(4)}

Notes

As a point on the Euler line, X₄₀₃ has Shinagawa coefficients \((2F, 2F - E)\).

X₄₀₃ is centroid of the triangle having vertices X₄, P(4), U(4). (Regarding the bicentric pair P(4) and U(4), see the notes just before X₁₉₀₈). (Randy Hutson, 9/23/2011)

Let A' be the reflection in BC of the A-vertex of the tangential triangle, and define B' and C' cyclically. The circumcircles of AB'C', BC'A', and CA'B' concur at the isogonal conjugate of X₄₀₃. Also, X₄₀₃ is the perspector of ABC and the reflection of the anticevian triangle of X₄ in the orthic axis (trilinear polar of X₄). (Randy Hutson, July 23, 2015)