AoPS - Problem

Source: Indonesian MO (INAMO) 2020, Day 1, Problem 1

Tags: geometry, circles, 2020, bisector, Indonesia MO, Indonesian MO

13.10.2020 11:00

Since this is already 3 PM (GMT +7) in Jakarta, might as well post the problem here. Problem 1. Given an acute triangle $ABC$ and the point $D$ on segment $BC$. Circle $c_1$ passes through $A, D$ and its centre lies on $AC$. Whereas circle $c_2$ passes through $A, D$ and its centre lies on $AB$. Let $P \neq A$ be the intersection of $c_1$ with $AB$ and $Q \neq A$ be the intersection of $c_2$ with $AC$. Prove that $AD$ bisects $\angle{PDQ}$.

Let the perpendicular from $D$ wrt $AD$ intersect $AB$ and $AC$ at $K$ and $J$, respectively. Since $ \measuredangle ADK = \measuredangle ADJ = 90^{\circ} $, we must have $AK$ and $AJ$ as the diameters of $(ADK)$ and $ADJ$, respectively, observe that $$ \measuredangle KQJ= \measuredangle AQK = \measuredangle ADK = \measuredangle ADJ = \measuredangle APJ = \measuredangle KPJ =90^{\circ} $$This gives $PKJQ$ is a cyclic quadrilateral, thus $$\measuredangle ADP =\measuredangle AJP = \measuredangle QJP = \measuredangle QKP = \measuredangle QKA = \measuredangle QDA $$As required.