X₁₄₇₈

CENTER OF JOHNSON-YFF CIRCLE

The Johnson Circle Theorem is the fact that if three congruent circles intersect in a point, then the circle passing through the other three intersections is congruent to them. This fourth circle is the Johnson circle of the three given circles. There are three congruent circles each tangent to two sides of triangle ABC. Peter Yff proved that their Johnson circle has center X₁₄₇₈. The circle is here named the Johnson-Yff circle of the triangle.

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

X₁₄₇₈ (CENTER OF JOHNSON-YFF CIRCLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 4}, {3, 12}

• Related Centers: X₃₄₂₂ (-VERTEX CONJUGATE OF X₁₉), X₁ (INCENTER), X₄ (ORTHOCENTER), X₃ (CIRCUMCENTER), X₁₂ ({X)

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

Barycentrics: \(a^{4} + 2 a^{2} b c - (b^{2} - c^{2})^{2} : :\)

Search Value (6, 9, 13): \(0.322897294537\)

▸ Linear Combinations2 combinations

• X₁₄₇₈ = \((R/r)\) X₁ + \(3\) X₂ - \(2\) X₃

• X₁₄₇₈ = \(R\) X₁ - \(r\) X₃ + \(2r\) X₅

▸ Incident Central Lines23 lines

{1, 4}, {2, 36}, {3, 12}, {5, 56}, {7, 80}, {8, 79}, {10, 46}, {11, 381}, {13, 203}, {14, 202}, {30, 55}, {65, 68}, {119, 1470}, {148, 192}, {442, 958}, {474, 1329}, {496, 546}, {529, 956}, {612, 1370}, {908, 997}, {975, 1076}, {990, 1074}, {1352, 1469}

▸ Functional Relations9 relations

• X₁₄₇₈ = reflection of X_i in X_j for these (i,j): (1, 226), (55, 495), (63, 10)

• X₁₄₇₈ = isogonal conjugate of X₃₄₂₂

• X₁₄₇₈ = anticomplement of X₉₉₃

• X₁₄₇₈ = X₁₀₆₅-Ceva conjugate of X₁

• X₁₄₇₈ = homothetic center of 2nd isogonal triangle of X₁ and anticomplementary triangle; see X₃₆

• X₁₄₇₈ = homothetic center of anticomplementary triangle and cross-triangle of ABC and 2nd isogonal triangle of X₁

• X₁₄₇₈ = homothetic center of 2nd isogonal triangle of X₁ and cross-triangle of ABC and 2nd isogonal triangle of X₁

• X₁₄₇₈ = homothetic center of Johnson triangle and cross-triangle of ABC and 1st Johnson-Yff triangle

• X₁₄₇₈ = homothetic center of Ehrmann mid-triangle and 2nd Johnson-Yff triangle

Notes

Roger A. Johnson, Advanced Euclidean Geometry, Dover, New York, 1960, page 75.

Peter Yff, “Three concurrent congruent circles ‘inscribed’ in a triangle,” manuscript, 1998; X₁₄₇₈ is the point C on page 5. See also X₄₉₅-X₄₉₉.