X3513

1st DILATION CENTER

For any two points P, Q in the set {A, B, C} and arbitrary point X, let M(X) = (|PX| + |QX|)/|PQ|. The point X that minimizes M(X) is X3513. See
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 3514, 3660 }: OI Line (Incenter–Circumcenter Line) { 1, 3 }: OI Line (Incenter–Circumcenter Line) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3514 = INVERSE-IN-CIRCUMCIRCLE OF X(3513) (1.1372, 1.2727) X_3660 = MIDPOINT OF {X(3513), X(3514)} (0.8653, 0.7732) X_1 = INCENTER (0.5714, 0.2333) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_3513 = 1st DILATION CENTER (0.5934, 0.2737) A B C X3514 X3660 X1 X3 X3513 (a, b, c) = (5, 6, 7) | AB = 1

X3513 (1st DILATION CENTER) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \((a - b + c)(a + b - c) + 4r(r^{2} + 4rR)^{1/2} : :\) (Peter Moses, 7/22/08)
Tripolars: \(1/r_{a} : :\), where \(r_{a}\), \(r_{b}\), \(r_{c}\) are the exradii
Search Value (6, 9, 13): \(0.847390686637\)
▸ Alternate Coordinates2 forms

  • Tripolars: \(\tan A' : :\), where JaJbJc is the excentral triangle

  • Tripolars: \(b + c - a : :\)

▸ Linear Combinations1 combination

  • X3513 = \((r + 2R + \sqrt{r^{2} + 4rR})\) X1 - \(2r\) X3

▸ Incident Central Lines1 line

{1, 3}

▸ Functional Relations5 relations

Notes

David Eppstein, The dilation center of a triangle

for a discussion of X3513 as a point of intersection of three similar ellipses, each having two of the points A, B, C as foci. (Contributed by David Eppstein, 7/10/08)

X3513 is the point P such that the incenter of the circumcevian triangle of P is also the incenter of triangle ABC. (Randy Hutson, 9/23/2011)

X3513 and X3514 are the limiting points (point-circles) of the coaxal system that includes the circumcircle and incircle. Their midpoint, X3660, is the radical trace of the incircle and circumcircle. (Peter Moses, November 15, 2011)

In the plane of a triangle ABC, let P be a point on the line X1X3, and let DEF = intouch triangle. Let \(R_{A}\) = reflection of AP in EF, and define \(R_{B}\) and \(R_{C}\) cyclically. The lines \(R_{A}\), \(R_{B}\), \(R_{C}\) concur in a point, P', that lies on a rectangular hyperbola that circumscribes DEF. Let P″ = DEF-isogonal conjugate of P'. The mapping P → P″ is a projective transformation of the line X1X3, and its fixed points are X3513 and X3514. (Angel Montesdeoca, April 17, 2021)