MathDB

Euler Line perpendicular BI criterion in isosceles (2020 Kyiv City MO 10.5 11.4)

Source:

September 20, 2020

geometryEuler Lineincentercollinearperpendicularisosceles

Problem Statement

Given an acute isosceles triangle

ABC, AK

and

CN

are its angle bisectors,

I

is their intersection point . Let point

X

be the other intersection point of the circles circumscribed around

\vartriangle ABC

and

\vartriangle KBN

. Let

M

be the midpoint of

AC

. Prove that the Euler line of

\vartriangle ABC

is perpendicular to the line

BI

if and only if the points

X, I

and

M

lie on the same line.
(Kivva Bogdan)