Alexei Myakishev, On the procircumcenter and related points,
Forum Geometricorum 3 (2003) 29--34.

Abstract: Given a triangle ABC, we solve the construction problem of a point P, together with points B_c, C_b on BC, C_a,  A_c on CA, and A_b, B_a on AB such that PB_aC_a, A_bPC_b, and A_cB_cP are congruent triangles similar to ABC. There are  altogether seven such triads.  If these three congruent triangles are all oppositely similar to ABC, then P must be the procircumcenter, with trilinear coordinates (a^2 cos A : b^2 cos B : c^2cos C). If at least one of the  triangles in the triad is directly similar to ABC, then P is  either a vertex or the midpoint of a side of the tangential triangle.  We also determine the ratio of similarity in each case.

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