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Omni-MATH / Let ABCD be a quadrilateral inscribed in a unit circle with center O. Suppose that ∠AOB=∠COD=135^(∘),BC=1. Let B^{prime} and C^{prime} be the reflections of A across BO and CO resp…

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problem

Let ABCDA B C D be a quadrilateral inscribed in a unit circle with center OO. Suppose that ∠AOB=∠COD=135∘,BC=1\angle A O B=\angle C O D=135^{\circ}, B C=1. Let $B^{\prime}$ and $C^{\prime}$ be the reflections of AA across BOB O and COC O respectively. Let H1H_{1} and H2H_{2} be the orthocenters of $A B^{\prime} C^{\prime}$ and BCDB C D, respectively. If MM is the midpoint of OH1O H_{1}, and $O^{\prime}$ is the reflection of OO about the midpoint of MH2M H_{2}, compute $O O^{\prime}$.
Plain-text mathematical notation (without MathML)
Let ABCD be a quadrilateral inscribed in a unit circle with center O. Suppose that ∠AOB=∠COD=135^(∘),BC=1. Let $B^{\prime}$ and $C^{\prime}$ be the reflections of A across BO and CO respectively. Let H₁ and H₂ be the orthocenters of $A B^{\prime} C^{\prime}$ and BCD, respectively. If M is the midpoint of OH₁, and $O^{\prime}$ is the reflection of O about the midpoint of MH₂, compute $O O^{\prime}$.
Original LaTeX notation
Let $A B C D$ be a quadrilateral inscribed in a unit circle with center $O$. Suppose that $\angle A O B=\angle C O D=135^{\circ}, B C=1$. Let $B^{\prime}$ and $C^{\prime}$ be the reflections of $A$ across $B O$ and $C O$ respectively. Let $H_{1}$ and $H_{2}$ be the orthocenters of $A B^{\prime} C^{\prime}$ and $B C D$, respectively. If $M$ is the midpoint of $O H_{1}$, and $O^{\prime}$ is the reflection of $O$ about the midpoint of $M H_{2}$, compute $O O^{\prime}$.

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Source and history

task_id
21aa8ee7-26ab-5d88-ba7a-7910b024413a
task_key
test--21aa8ee7-26ab-5d88-ba7a-7910b024413a
task_revision_id
1
config / split
/ test

Official source

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