Olym 10G. 사영 기하학 심화
- 실전문제
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Let $AXBY$ be a convex quadrilateral. The incircle of $\triangle AXY$ has center $I_A$ and touches $\overline{AX}$ and $\overline{AY}$ at $A_1$ and $A_2$ respectively. The incircle of $\triangle BXY$ has center $I_B$ and touches $\overline{BX}$ and $\overline{BY}$ at $B_1$ and $B_2$ respectively. Define $P = \overline{XI_A} \cap \overline{YI_B}$, $Q = \overline{XI_B} \cap \overline{YI_A}$, and $R = \overline{A_1B_1} \cap \overline{A_2B_2}$.
Prove that if $\angle AXB = \angle AYB$, then $P$, $Q$, $R$ are collinear.
Prove that if there exists a circle tangent to all four sides of $AXBY$, then $P$, $Q$, $R$ are collinear.
USMCA 2019
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