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Omni-MATH / Given circle O with radius R, the inscribed triangle ABC is an acute scalene triangle, where AB is the largest side. AH_(A),BH_(B),CH_(C) are heights on BC,CA,AB. Let D be the symm…

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problem

Given circle OO with radius RR, the inscribed triangle ABCABC is an acute scalene triangle, where ABAB is the largest side. AHA,BHB,CHCAH_A, BH_B,CH_C are heights on BC,CA,ABBC,CA,AB. Let DD be the symmetric point of HAH_A with respect to HBHCH_BH_C, EE be the symmetric point of HBH_B with respect to HAHCH_AH_C. PP is the intersection of AD,BEAD,BE, HH is the orthocentre of △ABC\triangle ABC. Prove: OP⋅OHOP\cdot OH is fixed, and find this value in terms of RR. (Edited)
Plain-text mathematical notation (without MathML)
Given circle O with radius R, the inscribed triangle ABC is an acute scalene triangle, where AB is the largest side. AH_(A),BH_(B),CH_(C) are heights on BC,CA,AB. Let D be the symmetric point of H_(A) with respect to H_(B)H_(C), E be the symmetric point of H_(B) with respect to H_(A)H_(C). P is the intersection of AD,BE, H is the orthocentre of △ABC. Prove: OP⋅OH is fixed, and find this value in terms of R.

(Edited)
Original LaTeX notation
Given circle $O$ with radius $R$, the inscribed triangle $ABC$ is an acute scalene triangle, where $AB$ is the largest side. $AH_A, BH_B,CH_C$ are heights on $BC,CA,AB$. Let $D$ be the symmetric point of $H_A$ with respect to $H_BH_C$, $E$ be the symmetric point of $H_B$ with respect to $H_AH_C$. $P$ is the intersection of $AD,BE$, $H$ is the orthocentre of $\triangle ABC$. Prove: $OP\cdot OH$ is fixed, and find this value in terms of $R$.

(Edited)

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task_id
9e3ceaa4-f3ad-5ce5-9a75-3807958f0be0
task_key
test--9e3ceaa4-f3ad-5ce5-9a75-3807958f0be0
task_revision_id
1
config / split
/ test

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