Let I_{a}(r_{a}), I_{b}(r_{b}), I_{c}(r_{c}) be excircles of the ABC. There are four circles tangent to I_{b}(r_{b}), I_{c}(r_{c}) and passing through I_{a}.

Let (S_a) the circle touching I_{b}(r_{b}), I_{c}(r_{c}) externally, and d_{a} the line passing through the points of contact of (S_a) with I_{b}(r_{b}) and I_{c}(r_{c})
Similary construct d_{b} and d_{c}. Let A_1B_1C_! be the triangle bounded by the lines d_{a}, d_{b} and d_{c}. Then the triangles ABC and A_1B_1C_1 are perspective.
Let (T_a) the circle touching I_{b}(r_{b}), I_{c}(r_{c}) internally, and d_{a} the line passing through the points of contact of (T_a) with I_{b}(r_{b}) and I_{c}(r_{c})
Similary construct d_{b} and d_{c}. Let A_2B_2C_2 be the triangle bounded by the lines d_{a}, d_{b} and d_{c}. Then the triangles ABC and A_2B_2C_2 are perspective.
Angel Montesdeoca