MathDB

Perpendicularity implies special intersection

Source: Germany VAIMO 2015 - #5

July 11, 2015

geometrycircumcircleincenterperpendicularorthogonal

Problem Statement

Let

ABC

be an acute triangle with the circumcircle

k

and incenter

I

. The perpendicular through

I

in

CI

intersects segment

[BC]

in

U

and

X

in

XI

. In particular

V

and

A

are on different sides of

BC

. The parallel line through

U

to

AI

intersects

AV

in

X

. Prove: If

XI

and

AI

are perpendicular to each other, then

XI

intersects segment

[AC]

in its midpoint

M

.
(Notation:

[\cdot]

denotes the line segment.)

Back to Problems View on AoPS