X₁₃₂₃

FLETCHER POINT

X₁₃₂₃ is the point of intersection of the line X₁X₇ and the trilinear polar of X₇. These two lines are orthogonal.

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

X₁₃₂₃ (FLETCHER POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1155, 3328}

• Related Centers: X₁₁₅₅ (SCHRÖDER POINT), X₃₃₂₈ (INCIRCLE TRANSFORM T(X₂₂₉₁))

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = (\sec^{2} A/2)(2 \cos^{2} A/2 - \cos^{2} B/2 - \cos^{2} C/2)\)

Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)

Search Value (6, 9, 13): \(-0.663746772240\)

▸ Alternate Coordinates1 form

• Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = (2a^{2} - b^{2} - c^{2} - ab - ac + 2bc)/(b + c - a) [P\). J. C. Moses, 6/25/04]

▸ Incident Central Lines10 lines

{1, 7}, {10, 348}, {36, 934}, {40, 738}, {85, 1125}, {106, 927}, {165, 479}, {241, 514}, {519, 664}, {1319, 1355}

▸ Functional Relations6 relations

• X₁₃₂₃ = midpoint of X₁₁₅₅ and X₃₃₂₈

• X₁₃₂₃ = isogonal conjugate of X₄₈₄₅

• X₁₃₂₃ = inverse-in-incircle of X₇

• X₁₃₂₃ = X₁₂₆₀-cross conjugate of X₅₂₇

• X₁₃₂₃ = crossdifference of every pair of points on line X₅₅X₆₅₇

• X₁₃₂₃ = X₁₈₇-of-intouch-triangle

Notes

X₁₃₂₃ is named in honor of T. J. Fletcher in

Adrian Oldknow, “The Euler-Gergonne-Soddy Triangle of a Triangle,” American Mathematical Monthly 103 (1996) 319-329.

See also MathWorld, Fletcher Point

X₁₃₂₃ is the radical trace of the inner and outer Soddy circles.