AoPS - Problem

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Problem

Source: 2020 Greek TST p2

Tags: geometry, concurrency, concurrent

parmenides51

14.11.2020 22:56

Given a triangle $ABC$ inscribed in circle $c(O,R)$ (with center $O$ and radius $R$ ) with $AB<AC<BC$ and let $BD$ be a diameter of the circle $c$ . The perpendicular bisector of $BD$ intersects line $AC$ at point $M$ and line $AB$ at point $N$ . Line $ND$ intersects the circle $c$ at point $T$ . Let $S$ be the second intersection point of cicumcircles $c_1$ of triangle $OCM$ , and $c_2$ of triangle $OAD$ . Prove that lines $AD, CT$ and $OS$ pass through the same point.

Com10atorics

18.11.2020 19:35

parmenides51 wrote: and line $AB$ at point $M$ . . It should be $N$ . Now, $\angle NOD=90=\angle NAD$ gives that $DOAN$ is cyclic. $\angle DTO=\angle NDB=\angle NBO$ gives that $NBOT$ is cyclic. And $\angle TOM=\angle TON=\angle TBN=\angle TBA=\angle TCA=\angle TCM$ so $TMOC$ is also cyclic and by Radical Axis Theorem in $c$ , $c_1$ and $c_2$ we are done.