MathDB

intersections of the angle bisector with intouch triangle

Source: Middle European Mathematical Olympiad T-5

September 21, 2014

geometryincentercircumcircleangle bisectorperpendicular bisectorgeometry proposed

Problem Statement

Let

ABC

be a triangle with

AB < AC

. Its incircle with centre

I

touches the sides

BC, CA,

and

AB

in the points

D, E,

and

F

respectively. The angle bisector

AI

intersects the lines

DE

and

DF

in the points

X

and

Y

respectively. Let

Z

be the foot of the altitude through

A

with respect to

BC

.
Prove that

D

is the incentre of the triangle

XYZ