X₁₁₅₅

SCHRÖDER POINT

Let XYZ be the intouch triangle of ABC; i.e., the pedal triangle of the incenter, I. The circles AIX, BIY, CIZ concur in two points. One of them is I; the other is X₁₁₅₅. This result is obtain by inversion in

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

X₁₁₅₅ (SCHRÖDER POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1319, 36}

• Related Centers: X₁₃₁₉ (BEVAN-SCHRÖDER POINT), X₃₆ (INVERSE-IN-CIRCUMCIRCLE OF INCENTER)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = \cos B + \cos C - 2 \cos A\);

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

Search Value (6, 9, 13): \(-4.708084434718\)

▸ Alternate Coordinates2 forms

• Trilinears: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = (b - c)^{2} + a(b + c - 2a)\)

• Barycentrics: \(ag(a,b,c) : bg(b,c,a) : cg(c,a,b)\)

X₁₁₅₅ lies on the Darboux quintic.

▸ Incident Central Lines16 lines

{1, 3}, {10, 535}, {11, 516}, {37, 750}, {44, 513}, {47, 582}, {63, 210}, {88, 105}, {89, 1002}, {100, 518}, {227, 603}, {238, 1054}, {243, 653}, {244, 902}, {377, 667}, {404, 960}

▸ Functional Relations17 relations

• X₁₁₅₅ = midpoint of X_i and X_j for these (i,j): (1, 3245), (36, 484), (32622, 32623)

• X₁₁₅₅ = reflection of X₁₃₁₉ in X₃₆

• X₁₁₅₅ = isogonal conjugate of X₁₁₅₆

• X₁₁₅₅ = inverse-in-circumcircle of X₅₅

• X₁₁₅₅ = inverse-in-incircle of X₃₅₄

• X₁₁₅₅ = inverse-in-Bevan-circle of X₅₇

• X₁₁₅₅ = crosspoint of X_i and X_j for these (i,j): (1, 1156), (527, 1323)

• X₁₁₅₅ = crosssum of X₁ and X₁₁₅₅

• X₁₁₅₅ = crossdifference of every pair of points on line X₁X₆₅₀

• X₁₁₅₅ = complement of X₅₀₅₇

• X₁₁₅₅ = anticomplement of X₅₀₈₇

• X₁₁₅₅ = orthogonal projection of X₁ on its trilinear polar

• X₁₁₅₅ = inverse-in-{circumcircle, incircle}-inverter of X₁

• X₁₁₅₅ = homothetic center of intouch triangle and medial triangle of 1st circumperp triangle

• X₁₁₅₅ = endo-homothetic center of X₂- and X₄-Ehrmann triangles; the homothetic center is X₈₅₈

• X₁₁₅₅ = X₄₆₈-of-excentral-triangle

• X₁₁₅₅ = {X₁,X₃}-harmonic conjugate of X₃₇₆₀₀

Notes

Heinz Schröder, “Die Inversion und ihre Anwendung im Unterricht der Oberstufe,” Der Mathematikunterricht 1 (1957) 59-80.

Each vertex of the tangential triangle of any triangle T is the inverse-in-the-circumcircle-of-T of the midpoints of the sides of T. Applying this to triangle XYZ shows that X₁₁₅₅ is the inverse-in-the-incircle of the centroid of XYZ; i.e., X₁₁₅₅ is X₂₃-of-the-intouch-triangle. (Darij Grinberg, #6319, 1/11/03; coordinates by Jean-Pierre Ehrmann, #6320, 1/11/03)