HYACINTHOS 28871

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and A'B'C' the pedal triangle of I.

Denote:

(Oa), (Ob), (Oc) = the Soddy circles (A, AB'=AC'), (B, BC'=BA'), (C, CA'=CB'), resp.

1. The circle tangent to (Oa), (Ob), (Oc) internally touches them at A", B", C", resp.

The quadrangles B'C'C"B", C'A'A"C", A'B'B"A" are cyclic (*)

(*) Carlos Hugo Olivera Díaz, PERU GEOMETRICO

 

Let Wa, Wb, Wc be the centers of the circles (B'C'C"B"), (C'A'A"C"), (A'B'B"A"), resp..

A'B'C', WaWbWc are perspective and orthologic.

Note: If ABC is not acute angled we take the largest circle tangent externally the circles (Oa), (Ob), (Oc)

 

2. The circle tangent to (Oa), (Ob), (Oc) externally touches them at A*, B*, C*, resp.

The quadrangles B'C'C*B*, C'A'A*C*, A'B'B*A* are cyclic 

 Let W1, W2, W3 be the centers of the circles (B'C'C*B*), (C'A'A*C*), (A'B'B*A*), resp..

A'B'C', W1W2W3 are perspective and orthologic.

 

APH