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Collinear iff Concyclic

Source: KJMO 2021 P4

November 13, 2021

geometryangle bisectorperpendicular bisectorcircumcircle

Problem Statement

In an acute triangle

ABC

with

\overline{AB} < \overline{AC}

, angle bisector of

A

and perpendicular bisector of

\overline{BC}

intersect at

D

. Let

P

be an interior point of triangle

ABC

. Line

CP

meets the circumcircle of triangle

ABP

again at

K

. Prove that

B, D, K

are collinear if and only if

AD

and

BC

meet on the circumcircle of triangle

APC

.

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