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Cute geometry problem

Source: 2023 4th OMpD L2 P4 - Brazil - Olimpíada Matemáticos por Diversão

September 21, 2023

geometrycircumcircletangenttangent circles

Problem Statement

Let

ABC

be a scalene acute triangle with circumcenter

O

. Let

K

be a point on the side

\overline{BC}

. Define

M

as the second intersection of

\overleftrightarrow{OK}

with the circumcircle of

BOC

. Let

L

be the reflection of

K

by

\overleftrightarrow{AC}

. Show that the circumcircles of the triangles

LCM

and

ABC

are tangent if, and only if,

\overline{AK} \perp \overline{BC}

.

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