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Omni-MATH / Let triangle ABC have incircle ω, which touches BC,CA, and AB at D,E, and F, respectively. Then, let ω₁ and ω₂ be circles tangent to AD and internally tangent to ω at E and F, resp…

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problem

Let triangle ABCABC have incircle ω\omega, which touches BC,CABC, CA, and ABAB at D,ED, E, and FF, respectively. Then, let ω1\omega_{1} and ω2\omega_{2} be circles tangent to ADAD and internally tangent to ω\omega at EE and FF, respectively. Let PP be the intersection of line EFEF and the line passing through the centers of ω1\omega_{1} and ω2\omega_{2}. If ω1\omega_{1} and ω2\omega_{2} have radii 5 and 6, respectively, compute PE⋅PFPE \cdot PF.
Plain-text mathematical notation (without MathML)
Let triangle ABC have incircle ω, which touches BC,CA, and AB at D,E, and F, respectively. Then, let ω₁ and ω₂ be circles tangent to AD and internally tangent to ω at E and F, respectively. Let P be the intersection of line EF and the line passing through the centers of ω₁ and ω₂. If ω₁ and ω₂ have radii 5 and 6, respectively, compute PE⋅PF.
Original LaTeX notation
Let triangle $ABC$ have incircle $\omega$, which touches $BC, CA$, and $AB$ at $D, E$, and $F$, respectively. Then, let $\omega_{1}$ and $\omega_{2}$ be circles tangent to $AD$ and internally tangent to $\omega$ at $E$ and $F$, respectively. Let $P$ be the intersection of line $EF$ and the line passing through the centers of $\omega_{1}$ and $\omega_{2}$. If $\omega_{1}$ and $\omega_{2}$ have radii 5 and 6, respectively, compute $PE \cdot PF$.

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Source and history

task_id
58b4f436-4189-5e41-a243-e323870a54a7
task_key
test--58b4f436-4189-5e41-a243-e323870a54a7
task_revision_id
1
config / split
/ test

Official source

initial import