Let $ABC$ be an acute scalene triangle with circumcenter $O$, and let $T$ be on line $BC$ such that $\angle TAO = 90^{\circ}$. The circle with diameter $\overline{AT}$ intersects the circumcircle of $\triangle BOC$ at two points $A_1$ and $A_2$, where $OA_1 < OA_2$. Points $B_1$, $B_2$, $C_1$, $C_2$ are defined analogously.
Prove that $\overline{AA_1}$, $\overline{BB_1}$, $\overline{CC_1}$ are concurrent.
Prove that $\overline{AA_2}$, $\overline{BB_2}$, $\overline{CC_2}$ are concurrent on the Euler line of triangle $ABC$.
USA TST 2017
- ahmedittihad
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Frankly, my dear, I don't give a damn.
- Raiyan Jamil
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Re: USA TST 2017
$\text{Solution of (1):}$
$\text{Solution of (2):}$
A smile is the best way to get through a tough situation, even if it's a fake smile.
Re: USA TST 2017
Hints for the solution.
First part:
Second part:
This was a rather bland problem for a TST.
First part: