IMO Hk TST

Geometry Difficulty 8.5 Shortlist Prove it Hong Kong

$\triangle ABC$ is an acute triangle. Let $A_1$ be the centre of the square inscribed in $\triangle ABC$ having two vertices on side $BC$ . Let $B_1$ be the centre of the square inscribed in $\triangle ABC$ having two vertices on side $CA$ . Let $C_1$ be the centre of the square inscribed in $\triangle ABC$ having two vertices on side $AB$ . Prove that lines $AA_1$ , $BB_1$ and $CC_1$ are concurrent.

Solution

Let $DEFG$ be the square with $A_1$ as centre such that $D, E$ lie on $BC$ , $F$ lies on $CA$ , and $G$ lies on $AB$ . By considering a homothety with centre $A$ , we can map $DEFG$ to a square $D'E'CB$ since $GF // BC$ . Let $X$ be the centre of $D'E'CB$ . Due to the homothety, $A$ , $A_1$ , $X$ are collinear. Similarly, let $Y$ and $Z$ be the centres of the squares constructed outside $\triangle ABC$ having $CA$ and $AB$ as a side respectively. It suffices to prove $AX$ , $BY$ , $CZ$ are concurrent. This follows from Jacobi's theorem since $\angle ZAB = \angle YAC = 45^\circ$ , $\angle XBC = \angle ZBA = 45^\circ$ and $\angle YCA = \angle XCB = 45^\circ$ .

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