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Problem 4 -- Problematic Parallel and Perpendicular Proofs

Source: 46th Austrian Mathematical Olympiad Regional Competition Problem 4

July 14, 2018

Austriageometry

Problem Statement

Let

ABC

be an isosceles triangle with

AC = BC

and

\angle ACB < 60^\circ

. We denote the incenter and circumcenter by

I

and

O

, respectively. The circumcircle of triangle

BIO

intersects the leg

BC

also at point

D \ne B

.
(a) Prove that the lines

AC

and

DI

are parallel. (b) Prove that the lines

OD

and

IB

are mutually perpendicular.
(Walther Janous)