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Omni-MATH / In acute triangle ABC, let H be the orthocenter and D the foot of the altitude from A. The circumcircle of triangle BHC intersects AC at E≠C, and AB at F≠B. If BD=3,CD=7, and …

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problem

In acute triangle ABCABC, let HH be the orthocenter and DD the foot of the altitude from AA. The circumcircle of triangle BHCBHC intersects ACAC at E≠CE \neq C, and ABAB at F≠BF \neq B. If BD=3,CD=7BD=3, CD=7, and AHHD=57\frac{AH}{HD}=\frac{5}{7}, the area of triangle AEFAEF can be expressed as ab\frac{a}{b}, where a,ba, b are relatively prime positive integers. Compute 100a+b100a+b.
Plain-text mathematical notation (without MathML)
In acute triangle ABC, let H be the orthocenter and D the foot of the altitude from A. The circumcircle of triangle BHC intersects AC at E≠C, and AB at F≠B. If BD=3,CD=7, and (AH)/(HD)=(5)/(7), the area of triangle AEF can be expressed as (a)/(b), where a,b are relatively prime positive integers. Compute 100a+b.
Original LaTeX notation
In acute triangle $ABC$, let $H$ be the orthocenter and $D$ the foot of the altitude from $A$. The circumcircle of triangle $BHC$ intersects $AC$ at $E \neq C$, and $AB$ at $F \neq B$. If $BD=3, CD=7$, and $\frac{AH}{HD}=\frac{5}{7}$, the area of triangle $AEF$ can be expressed as $\frac{a}{b}$, where $a, b$ are relatively prime positive integers. Compute $100a+b$.

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Source and history

task_id
09aec8e9-77e9-5119-8ef7-fbb1d5f3d44a
task_key
test--09aec8e9-77e9-5119-8ef7-fbb1d5f3d44a
task_revision_id
1
config / split
/ test

Official source

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