X₂₄₄

X₁-LINE CONJUGATE OF X₁₀₀

X₂₄₄ = crosssum of circumcircle intercepts of line X₁X₂₁

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

X₂₄₄ (X₁-LINE CONJUGATE OF X₁₀₀) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 38}, {31, 57}

• Related Centers: X₁ (INCENTER), X₁₀₀ (ANTICOMPLEMENT OF FEUERBACH POINT), X₂ (CENTROID), X₃₈ (CROSSPOINT OF X₁ AND X₇₅), X₃₁ (2nd POWER POINT), X₅₇ (ISOGONAL CONJUGATE OF X₉)

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

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

Search Value (6, 9, 13): \(1.157887174674\)

▸ Alternate Coordinates2 forms

• Trilinears: \([1 - \cos(B - C)]\sin^{2}(A/2) : :\)

• Trilinears: \(d(a,b,c) : :\), where d(a,b,c) = \((distance from A to Nagel line)^{2}\)

X₂₄₄ lies on aforementioned ellipses.

▸ Incident Central Lines13 lines

{1, 88}, {2, 38}, {11, 867}, {31, 57}, {34, 1106}, {42, 354}, {58, 229}, {63, 748}, {238, 896}, {474, 976}, {518, 899}, {596, 1089}, {665, 866}

▸ Functional Relations26 relations

• X₂₄₄ = isogonal conjugate of X₇₆₅

• X₂₄₄ = isotomic conjugate of X₇₀₃₅

• X₂₄₄ = anticomplement of X₂₄₀₀₃

• X₂₄₄ = crosssum of circumcircle intercepts of line X₁X₂₁

• X₂₄₄ = perspector of anti-Aquila and 2nd Sharygin triangles

• X₂₄₄ = barycentric product of vertices of Garcia reflection triangle

• X₂₄₄ = trilinear pole of line X₇₆₄X₂₀₈₇

• X₂₄₄ = X_i-Ceva conjugate of X_j for these (i,j): (1, 513), (75, 514)

• X₂₄₄ = crosspoint of X₁ and X₅₁₃

• X₂₄₄ = crosssum of X_i and X_j for these (i,j): (1, 100), (31, 101), (78, 1331), (109, 1420), (200, 644), (651, 1445), (678, 1023), (756, 1018)

• X₂₄₄ = crossdifference of every pair of points on line X₁₀₀X₁₀₁

• X₂₄₄ = X₁-Hirst inverse of X₁₀₅₄

• X₂₄₄ = X₁-line conjugate of X₁₀₀

• X₂₄₄ = complement of X₃₉₅₂

• X₂₄₄ = antipode of X₄₇₃₈ in inellipse centered at X₁₀

• X₂₄₄ = reflection of X₄₇₃₈ in X₁₀

• X₂₄₄ = bicentric difference of PU(34)

• X₂₄₄ = PU(34)-harmonic conjugate of X₁₆₃₅

• X₂₄₄ = tripolar centroid of X₁₀₂₂

• X₂₄₄ = perspector of circumconic centered at X₆₆₁

• X₂₄₄ = center of circumconic that is locus of trilinear poles of lines passing through X₆₆₁

• X₂₄₄ = X₂-Ceva conjugate of X₆₆₁

• X₂₄₄ = trilinear pole wrt incentral triangle of line X₁X₆

• X₂₄₄ = intersection of tangents to Steiner inellipse at X₁₀₁₅ and X₁₀₈₆

• X₂₄₄ = crosspoint wrt medial triangle of X₁₀₁₅ and X₁₀₈₆

• X₂₄₄ = trilinear square of X₅₁₃

Notes

Let O* be a circle with center X₃ and variable radius R*. Let L_a be the radical axis of O* and the A-excircle, and define L_b and L_c cyclically. Let A'=L_b∩L_c, B'=L_c∩L_a, C'=L_a∩L_b. Then A'B'C' is perspective to ABC, and the locus of the perspector as R* varies is the hyperbola {A, B, C,X₁,X₁₀}}, which has center X₂₄₄. Also, X₂₄₄ lies in the inellipse centered at X₁₀, as well as the Hofstadter ellpse E(1/2), which is the incentral inellipse. (Randy Hutson, December 26, 2016)

Art from triangle centers: Let X = X₂₄₄. In a square, suppose that P is a randomly chosen point, and form a triangle using two vertices of the square and P as the third vertex. Then let P = X and repeat the procedure, several thousand times. For typical results, see X₂₄₄ art (X(244)art.png) and X₂₄₄ art animated (X(244)KageyAnimated.gif). For results using other triangle centers for X, see X₄₁ art (X(41)art.png), X₅₈ art (X(58)art.png), X₇₆ art (X(76)art.png), X₁₁₅ art (X(115)art.png), X₁₄₁ art (X(141)art.png), X₁₇₉ art (X(179)art.png), and X₂₅₅ art (X(255)art.png). (Contributed by Peter Kagey, February 8, 2023) Figure 1 (KM)