A geometric result involving incentre and excentre

Question

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??

What is that 'circle'? Are you sure that they are 'touching'? — Mick, Mar 27 '20 at 08:21
@Mick: According to a GeoGebra sketch, OP probably means "The line $I_b A I_c$ meets the circumcircle of $\triangle ABC$ again at point $Q$." — Blue, Mar 27 '20 at 09:32

Mick · Accepted Answer · 2020-03-27T10:07:13.003

2

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 $\angle BAC$.

edited Mar 27 '20 at 10:07

answered Mar 27 '20 at 10:00

Mick

17,141

+1. Very nice! ... – Blue Mar 27 '20 at 10:09

A geometric result involving incentre and excentre

1 Answers1