P5

Part of 2024 Bangladesh Mathematical Olympiad

Problems(2)

Equal circumradii

Source: BdMO 2024 Secondary National P5

\triangle XPQ

\triangle YPQ

X

Y

are on the opposite sides of

PQ

and the circumradius of

\triangle XPQ

and the circumradius of

\triangle YPQ

I

J

are the incenters of

\triangle XPQ

\triangle YPQ

respectively. Let

M

be the midpoint of

PQ

I, M, J

are collinear. Prove that

XPYQ

is a parallelogram.

geometryincenterparallelogramAngle Chasing

Spot the symmetry

Source: BdMO 2024 Higher Secondary National P5

I

be the incenter of

\triangle ABC

P

be a point such that

PI

is perpendicular to

BC

PA

BC

. Let the line parallel to

BC

, which is tangent to the incircle of

\triangle ABC

AB

AC

Q

R

respectively. Prove that

\angle BPQ = \angle CPR

geometryincirclehomothetyAngle Chasingexcircle