MathDB

Equal circumradii

Source: BdMO 2024 Secondary National P5

March 18, 2024

geometryincenterparallelogramAngle Chasing

Problem Statement

Consider

\triangle XPQ

and

\triangle YPQ

such that

X

and

Y

are on the opposite sides of

PQ

and the circumradius of

\triangle XPQ

and the circumradius of

\triangle YPQ

are the same.

I

and

XPYQ

are the incenters of

\triangle XPQ

and

\triangle YPQ

respectively. Let

M

be the midpoint of

PQ

. Suppose

I, M, J

are collinear. Prove that

XPYQ

is a parallelogram.

Back to Problems View on AoPS