2024 AMC 8 Problems/Problem 15

Problem

Let $D$ be an interior point of the acute triangle $ABC$ with $AB > AC$ so that $\angle DAB= \angle CAD$ . The point $E$ on the segment $AC$ satisfies $\angle ADE= \angle BCD$ , the point $F$ on the segment $AB$ satisfies $\angle FDA= \angle DBC$ , and the point $X$ on the line $AC$ satisfies $CX=BX$ . Let $O_1$ and $O_2$ be the circumcentres of the triangles $ADC$ and $EXD$ respectively. Prove that the lines $BC$ , $EF$ , and $O_1 O_2$ are concurrent. (source: 2021 IMO)

now go do this problem as a punishment for trying to cheat

Solution

These are just left here for future conveniency.

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2024 AMC 8 Problems/Problem 15

Problem

Solution