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Omni-MATH / In an acute scalene triangle ABC, points D,E,F lie on sides BC,CA,AB, respectively, such that AD⊥BC,BE⊥CA,CF⊥AB. Altitudes AD,BE,CF meet at orthocenter H. Points P and Q lie on seg…

Problem

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problem

In an acute scalene triangle ABCABC, points D,E,FD,E,F lie on sides BC,CA,ABBC, CA, AB, respectively, such that AD⊥BC,BE⊥CA,CF⊥ABAD \perp BC, BE \perp CA, CF \perp AB. Altitudes AD,BE,CFAD, BE, CF meet at orthocenter HH. Points PP and QQ lie on segment EFEF such that AP⊥EFAP \perp EF and HQ⊥EFHQ \perp EF. Lines DPDP and QHQH intersect at point RR. Compute HQ/HRHQ/HR.
Plain-text mathematical notation (without MathML)
In an acute scalene triangle ABC, points D,E,F lie on sides BC,CA,AB, respectively, such that AD⊥BC,BE⊥CA,CF⊥AB. Altitudes AD,BE,CF meet at orthocenter H. Points P and Q lie on segment EF such that AP⊥EF and HQ⊥EF. Lines DP and QH intersect at point R. Compute HQ/HR.
Original LaTeX notation
In an acute scalene triangle $ABC$, points $D,E,F$ lie on sides $BC, CA, AB$, respectively, such that $AD \perp BC, BE \perp CA, CF \perp AB$. Altitudes $AD, BE, CF$ meet at orthocenter $H$. Points $P$ and $Q$ lie on segment $EF$ such that $AP \perp EF$ and $HQ \perp EF$. Lines $DP$ and $QH$ intersect at point $R$. Compute $HQ/HR$.

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Source and history

task_id
4b6fdd00-3538-5eb1-8cfa-fae84d2a258f
task_key
test--4b6fdd00-3538-5eb1-8cfa-fae84d2a258f
task_revision_id
3
config / split
/ test

Official source

initial import