A Concurrence Based on Ellipses through the Incenter

Let be the incenter of a triangle . A point satisfies the conditions , . The points and are defined analogously. Then the lines extended from , and intersect at the red point.

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References
[1] Colescu. "An Interesting Geometry Problem about Incenter and Ellipses." (Jan 9, 2020) math.stackexchange.com/questions/1897127/an-interesting-geometry-problem-about-incenter-and-ellipses.
[2] I. I. Bogdanov, "Two Theorems on the Focus-Sharing Ellipses: A Three-Dimensional View," Journal of Classical Geometry, 1, 2012 pp. 1–5. jcgeometry.org/Articles/Volume1/JCG2012V1pp1-5.pdf.
[3] S. Liebscher and D.-E. Liebscher, "The Love for the Three Conics," Forum Geometricorum, 18, 2018 pp. 419–430. forumgeom.fau.edu/FG2018volume18/FG201848.pdf.
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