what is the trilinear coordinates of the intersection between these 2 circles

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(I really searched on the internet, I found NOTHING)

In a triangle, we can draw the 9-point Euler circle, and the 3 excircles of that triangle. there always is an intersection point between the Euler circle, add one of the 3 excicles (unique point). I'd like to have the trilinear coordinates of these 3 points, because I searched everywhere in vain...