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Omni-MATH / Let A,B,C be points in that order along a line, such that AB=20 and BC=18. Let ω be a circle of nonzero radius centered at B, and let ℓ₁ and ℓ₂ be tangents to ω through A and C, re…

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problem

Let A,B,CA, B, C be points in that order along a line, such that AB=20A B=20 and BC=18B C=18. Let ω\omega be a circle of nonzero radius centered at BB, and let ℓ1\ell_{1} and ℓ2\ell_{2} be tangents to ω\omega through AA and CC, respectively. Let KK be the intersection of ℓ1\ell_{1} and ℓ2\ell_{2}. Let XX lie on segment KA¯\overline{K A} and YY lie on segment KC¯\overline{K C} such that XY‖BCX Y \| B C and XYX Y is tangent to ω\omega. What is the largest possible integer length for XYX Y?
Plain-text mathematical notation (without MathML)
Let A,B,C be points in that order along a line, such that AB=20 and BC=18. Let ω be a circle of nonzero radius centered at B, and let ℓ₁ and ℓ₂ be tangents to ω through A and C, respectively. Let K be the intersection of ℓ₁ and ℓ₂. Let X lie on segment (KA)¯ and Y lie on segment (KC)¯ such that XY‖BC and XY is tangent to ω. What is the largest possible integer length for XY?
Original LaTeX notation
Let $A, B, C$ be points in that order along a line, such that $A B=20$ and $B C=18$. Let $\omega$ be a circle of nonzero radius centered at $B$, and let $\ell_{1}$ and $\ell_{2}$ be tangents to $\omega$ through $A$ and $C$, respectively. Let $K$ be the intersection of $\ell_{1}$ and $\ell_{2}$. Let $X$ lie on segment $\overline{K A}$ and $Y$ lie on segment $\overline{K C}$ such that $X Y \| B C$ and $X Y$ is tangent to $\omega$. What is the largest possible integer length for $X Y$?

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task_id
6122f4fa-dd5a-5ff5-8a48-6e2d4520046a
task_key
test--6122f4fa-dd5a-5ff5-8a48-6e2d4520046a
task_revision_id
1
config / split
/ test

Official source

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