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Nice Quadrilateral Geo
amuthup   48
N 3 minutes ago by L13832
Source: 2021 ISL G4
Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.
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amuthup
Jul 12, 2022
L13832
3 minutes ago
A projectional vision in IGO
Shayan-TayefehIR   5
N an hour ago by egxa
Source: IGO 2024 Advanced Level - Problem 3
In the triangle $\bigtriangleup ABC$ let $D$ be the foot of the altitude from $A$ to the side $BC$ and $I$, $I_A$, $I_C$ be the incenter, $A$-excenter, and $C$-excenter, respectively. Denote by $P\neq B$ and $Q\neq D$ the other intersection points of the circle $\bigtriangleup BDI_C$ with the lines $BI$ and $DI_A$, respectively. Prove that $AP=AQ$.

Proposed Michal Jan'ik - Czech Republic
5 replies
Shayan-TayefehIR
Yesterday at 7:59 PM
egxa
an hour ago
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