Alexei Myakishev, On the procircumcenter and related
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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