X₁₀₀₁

MIDPOINT OF X₁ AND X₉

Let A* be the tangency point of O_A and the circumcircle, and define B* and C* cyclically. Let A** be the radical trace of O_B and O_C. The lines A*A**, B*B**, C*C** concur in X₁₀₀₁.

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

X₁₀₀₁ (MIDPOINT OF X₁ AND X₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 9}

• Related Centers: X₁ (INCENTER), X₉ (MITTENPUNKT)

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

Search Value (6, 9, 13): \(2.631140604343\)

▸ Linear Combinations1 combination

• X₁₀₀₁ = X₁ + \(3\) X₂ + \((r/R)\) X₃

▸ Incident Central Lines17 lines

{1, 6}, {2, 11}, {3, 142}, {7, 21}, {8, 344}, {31, 940}, {35, 474}, {42, 748}, {63, 354}, {182, 692}, {388, 452}, {527, 551}, {529, 1056}, {614, 968}, {750, 902}, {846, 982}, {943, 1058}

▸ Functional Relations12 relations

• X₁₀₀₁ = midpoint of X₁ and X₉

• X₁₀₀₁ = reflection of X₁₄₂ in X₁₁₂₅

• X₁₀₀₁ = isogonal conjugate of X₁₀₀₂

• X₁₀₀₁ = complement of X₂₅₅₀

• X₁₀₀₁ = crosssum of X_i and X_j for these (i,j): (116, 824), (788, 1015)

• X₁₀₀₁ = crossdifference of every pair of points on line X₅₁₃X₆₆₅

• X₁₀₀₁ = anticomplement of X₃₈₂₆

• X₁₀₀₁ = X₆-of-2nd-circumperp-triangle

• X₁₀₀₁ = X₁₄₁-of-hexyl-triangle

• X₁₀₀₁ = X₅₄₈₀-of-excentral-triangle

• X₁₀₀₁ = inverse-in-Feuerbach-hyperbola of X₅₅

• X₁₀₀₁ = polar conjugate of isotomic conjugate of X₂₃₁₅₁

Notes

Let O_A be the circle tangent of side BC at its midpoint and to the circumcircle on the side of BC opposite A. Define O_B and O_C cyclically. X₁₀₀₁ is the radical center of the circles O_A, O_B, O_C. (Randy Hutson, 9/23/2011)

Let L_A and M_A be the external tangents to circles O_B and O_C, with L_A being the farther from O_A. Define L_B, L_C, M_B, and M_C cyclically. Let A' = L_B∩L_C and A″ = M_B∩M_C, and define B', B″, C', and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X₁₀₀₁.

For details and relationships among X₁₀₀₁, X₁, X₁₄₅, X₃₃₆₁, X₃₆₁₆, X₃₉₁₃, and X₄₇₁₉, see

Luis González, “On a Triad of Circles Tangent to the Circumcircle and the Sides at Their Midpoints,” Forum Geometricorum 11 (2011) 145-154.

Let A' be the line through X₁ parallel to line BC. Let A_B = A'∩AB and A_C = A'∩AC. Define B_C and C_A cyclically, and define B_A and C_B cyclically. The six points A_B, B_C, C_A, A_B, B_C, C_A lie on a conic whose center is X₁₀₀₁. (Angel Montesdeoca, April 27, 2016)

X₁₀₀₁ is the {X₁,X₂₃₈}-harmonic conjugate of X₆.