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Omni-MATH / Let ABC be a fixed acute triangle inscribed in a circle ω with center O . A variable point X is chosen on minor arc AB of ω , and segments CX and AB meet at D . Denote by O₁ and O₂…

Problem

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problem

Let ABCABC be a fixed acute triangle inscribed in a circle ω\omega with center OO . A variable point XX is chosen on minor arc ABAB of ω\omega , and segments CXCX and ABAB meet at DD . Denote by O1O_1 and O2O_2 the circumcenters of triangles ADXADX and BDXBDX , respectively. Determine all points XX for which the area of triangle OO1O2OO_1O_2 is minimized.
Plain-text mathematical notation (without MathML)
Let ABC be a fixed acute triangle inscribed in a circle ω with center O . A variable point X is chosen on minor arc AB of ω , and segments CX and AB meet at D . Denote by O₁ and O₂ the circumcenters of triangles ADX and BDX , respectively. Determine all points X for which the area of triangle OO₁O₂ is minimized.
Original LaTeX notation
Let $ABC$ be a fixed acute triangle inscribed in a circle $\omega$ with center $O$ . A variable point $X$ is chosen on minor arc $AB$ of $\omega$ , and segments $CX$ and $AB$ meet at $D$ . Denote by $O_1$ and $O_2$ the circumcenters of triangles $ADX$ and $BDX$ , respectively. Determine all points $X$ for which the area of triangle $OO_1O_2$ is minimized.

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Source and history

task_id
0134743c-9694-50ae-a9e3-5bc775910948
task_key
test--0134743c-9694-50ae-a9e3-5bc775910948
task_revision_id
1
config / split
/ test

Official source

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