X₁₉₁₆

TRILINEAR POLE OF PU(11)

X₁₉₁₆ = midpoint of X₁₄₈ and X₁₉₄

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

X₁₉₁₆ (TRILINEAR POLE OF PU(11)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {148, 194}

• Related Centers: X₁₄₈ (STEINER POINT OF ANTICOMPLEMENTARY TRIANGLE), X₁₉₄ (-CEVA CONJUGATE OF X₂)

Trilinears: \(bc/(a^{4} - b^{2}c^{2}) : :\)

Search Value (6, 9, 13): \(-0.309770160722\)

▸ Alternate Coordinates1 form

• Trilinears: \(\csc(A - 2\omega ) : :\)

▸ Incident Central Lines15 lines

{2, 694}, {4, 147}, {10, 257}, {39, 83}, {76, 115}, {98, 385}, {114, 262}, {226, 335}, {256, 291}, {316, 736}, {325, 698}, {538, 671}, {543, 598}, {690, 882}, {804, 881}

▸ Functional Relations14 relations

• X₁₉₁₆ = midpoint of X₁₄₈ and X₁₉₄

• X₁₉₁₆ = reflection of X_i in X_j for these (i,j): (76, 115), (99, 39)

• X₁₉₁₆ = isogonal conjugate of X₁₆₉₁

• X₁₉₁₆ = isotomic conjugate of X₃₈₅

• X₁₉₁₆ = cevapoint of X₃₉ and X₅₁₁

• X₁₉₁₆ = complement of X₈₇₈₂

• X₁₉₁₆ = trilinear pole of line X₁₄₁X₅₂₃

• X₁₉₁₆ = pole wrt polar circle of trilinear polar of X₄₁₉

• X₁₉₁₆ = polar conjugate of isogonal conjugate of X₃₆₂₁₄

• X₁₉₁₆ = X₄₈-isoconjugate (polar conjugate) of X₄₁₉

• X₁₉₁₆ = antigonal image of X₇₆

• X₁₉₁₆ = X₇₆ of 1st anti-Brocard triangle

• X₁₉₁₆ = intersection, other than A, B, C, of the 1st and 2nd isobarycs of the circumcircle

• X₁₉₁₆ = perspector of ABC and 1st anti-Brocard triangle

Notes

Let A'B'C' be the 1st Brocard triangle. Let A″ be the reflection of A' in BC, and define B″ and C″ cyclically; then X₁₉₁₆ is the radical center of the circumcircles of A″BC, B″CA, C″AB. Let A* be the reflection of A in B'C', and define B* and C* cyclically; then X₁₉₁₆ is the radical center of the circumcircles of A*BC, B*CA, C*AB. The first set of circles equals the second set. (Randy Hutson, February 10, 2016)