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Vietnamese geometry in Kvant

Source: Kvant Magazine No. 10 2023 M2769

February 6, 2024

geometryincircleconcurrency

Problem Statement

The incircle of the triangle

ABC

touches the sides

BC, CA

and

AB{}

at

D,E

and

F{}

respectively. Let the circle

\omega

touch the segments

CA{}

and

AB{}

at

Q{}

and

R{}

respectively, and the points

M{}

and

N{}

are selected on the segments

AB{}

and

AC{}

respectively, so that the segments

CM{}

and

BN{}

touch

\omega

. The bisectors of

\angle NBC

and

\angle MCB

intersect the segments

DE{}

and

DF{}

at

K{}

and

L{}

respectively. Prove that the lines

RK{}

and

QL{}

intersect on

\omega

.
Proposed by Tran Quang Hung

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