Let Ia(ra), Ib(rb), Ic(rc) be the excircles of the ABC. There are four circles tangent to Ib(rb), Ic(rc) and passing through Ia.
Let (S_a) the circle touching Ib(rb), Ic(rc) externally, and pa the line passing through the points of contact of (S_a) with Ib(rb) and Ic(rc)
Similary construct pb and pc. Let A_1B_1C_1 be the triangle bounded by the lines da, db and dc. Then the triangles ABC and A_1B_1C_1 are perspective with prespector P ((6-9-13)-search number 3.471504569909629).
Let (T_a) the circle touching Ib(rb), Ic(rc) internally, and qa the line passing through the points of contact of (T_a) with Ib(rb) and Ic(rc)
Similary construct qb and qc. Let A_2B_2C_2 be the triangle bounded by the lines da, db and dc. Then the triangles ABC and A_2B_2C_2 are perspective with prespector Q ((6-9-13)-search number 0.2409473945412).
Angel Montesdeoca