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Problem

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problem

As shown in the following figure, a heart is a shape consist of three semicircles with diameters ABAB, BCBC and ACAC such that BB is midpoint of the segment ACAC. A heart ω\omega is given. Call a pair $(P, P')$ bisector if PP and $P'$ lie on ω\omega and bisect its perimeter. Let $(P, P')$ and $(Q,Q')$ be bisector pairs. Tangents at points $P, P', Q$, and $Q'$ to ω\omega construct a convex quadrilateral XYZTXYZT. If the quadrilateral XYZTXYZT is inscribed in a circle, find the angle between lines $PP'$ and $QQ'$. [img]https://cdn.artofproblemsolving.com/attachments/3/c/8216889594bbb504372d8cddfac73b9f56e74c.png[/img]
Plain-text mathematical notation (without MathML)
As shown in the following figure, a heart is a shape consist of three semicircles with diameters AB, BC and AC such that B is midpoint of the segment AC. A heart ω is given. Call a pair $(P, P')$ bisector if P and $P'$ lie on ω and bisect its perimeter. Let $(P, P')$ and $(Q,Q')$ be bisector pairs. Tangents at points $P, P', Q$, and $Q'$ to ω construct a convex quadrilateral XYZT. If the quadrilateral XYZT is inscribed in a circle, find the angle between lines $PP'$ and $QQ'$.
[img]https://cdn.artofproblemsolving.com/attachments/3/c/8216889594bbb504372d8cddfac73b9f56e74c.png[/img]
Original LaTeX notation
As shown in the following figure, a heart is a shape consist of three semicircles with diameters $AB$, $BC$ and $AC$ such that $B$ is midpoint of the segment $AC$. A heart $\omega$ is given. Call a pair $(P, P')$ bisector if $P$ and $P'$ lie on $\omega$ and bisect its perimeter. Let $(P, P')$ and $(Q,Q')$ be bisector pairs. Tangents at points $P, P', Q$, and $Q'$ to $\omega$ construct a convex quadrilateral $XYZT$. If the quadrilateral $XYZT$ is inscribed in a circle, find the angle between lines $PP'$ and $QQ'$.
[img]https://cdn.artofproblemsolving.com/attachments/3/c/8216889594bbb504372d8cddfac73b9f56e74c.png[/img]

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Source and history

task_id
97fefe32-10b9-511a-8479-313a784b4a71
task_key
test--97fefe32-10b9-511a-8479-313a784b4a71
task_revision_id
1
config / split
/ test

Official source

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