Let ABC is a triangle and $I_c$ and $I_b$ be excentres opposite to vertex C and vertex B respectively. Join A,$I_c$ and $I_b$. The line touches circle again at point Q .We have to prove that Q is the midpoint of line segment $I_bI_c$.I tried to prove but diagram got messy
another interesting result is circumdiameter through Q intersects circle at say P then P is the mid point of minor arc AB.Any help proving the results??
Consider the triangle formed by $I_a, I_b, I_c$. $AI_a$ is one of the altidues. The same is true for $BI_b, CI_c$.
Thus, the circum-circle is in fact the nine-point circle. It cuts that line at Q and Q by definitiion is the midpoint of $I_bI_c$.
The other fact is ture because $AI_a$ bisects $\angle BAC$.