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Omni-MATH / ABC is an acute triangle with incircle ω. ω is tangent to sides (BC)¯,(CA)¯, and (AB)¯ at D,E, and F respectively. P is a point on the altitude from A such that Γ, the circle with …

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problem

ABCA B C is an acute triangle with incircle ω\omega. ω\omega is tangent to sides BC¯,CA¯\overline{B C}, \overline{C A}, and AB¯\overline{A B} at D,ED, E, and FF respectively. PP is a point on the altitude from AA such that Γ\Gamma, the circle with diameter AP¯\overline{A P}, is tangent to ω\omega. Γ\Gamma intersects AC¯\overline{A C} and AB¯\overline{A B} at XX and YY respectively. Given XY=8,AE=15X Y=8, A E=15, and that the radius of Γ\Gamma is 5, compute BD⋅DCB D \cdot D C.
Plain-text mathematical notation (without MathML)
ABC is an acute triangle with incircle ω. ω is tangent to sides (BC)¯,(CA)¯, and (AB)¯ at D,E, and F respectively. P is a point on the altitude from A such that Γ, the circle with diameter (AP)¯, is tangent to ω. Γ intersects (AC)¯ and (AB)¯ at X and Y respectively. Given XY=8,AE=15, and that the radius of Γ is 5, compute BD⋅DC.
Original LaTeX notation
$A B C$ is an acute triangle with incircle $\omega$. $\omega$ is tangent to sides $\overline{B C}, \overline{C A}$, and $\overline{A B}$ at $D, E$, and $F$ respectively. $P$ is a point on the altitude from $A$ such that $\Gamma$, the circle with diameter $\overline{A P}$, is tangent to $\omega$. $\Gamma$ intersects $\overline{A C}$ and $\overline{A B}$ at $X$ and $Y$ respectively. Given $X Y=8, A E=15$, and that the radius of $\Gamma$ is 5, compute $B D \cdot D C$.

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task_id
2dcd5042-7ad2-59f4-8f44-3046a4c0d805
task_key
test--2dcd5042-7ad2-59f4-8f44-3046a4c0d805
task_revision_id
1
config / split
/ test

Official source

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