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Omni-MATH / Convex quadrilateral ABCD has right angles ∠A and ∠C and is such that AB=BC and AD=CD. The diagonals AC and BD intersect at point M. Points P and Q lie on the circumcircle of trian…

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problem

Convex quadrilateral ABCDA B C D has right angles ∠A\angle A and ∠C\angle C and is such that AB=BCA B=B C and AD=CDA D=C D. The diagonals ACA C and BDB D intersect at point MM. Points PP and QQ lie on the circumcircle of triangle AMBA M B and segment CDC D, respectively, such that points P,MP, M, and QQ are collinear. Suppose that m∠ABC=160∘m \angle A B C=160^{\circ} and m∠QMC=40∘m \angle Q M C=40^{\circ}. Find MP⋅MQM P \cdot M Q, given that MC=6M C=6.
Plain-text mathematical notation (without MathML)
Convex quadrilateral ABCD has right angles ∠A and ∠C and is such that AB=BC and AD=CD. The diagonals AC and BD intersect at point M. Points P and Q lie on the circumcircle of triangle AMB and segment CD, respectively, such that points P,M, and Q are collinear. Suppose that m∠ABC=160^(∘) and m∠QMC=40^(∘). Find MP⋅MQ, given that MC=6.
Original LaTeX notation
Convex quadrilateral $A B C D$ has right angles $\angle A$ and $\angle C$ and is such that $A B=B C$ and $A D=C D$. The diagonals $A C$ and $B D$ intersect at point $M$. Points $P$ and $Q$ lie on the circumcircle of triangle $A M B$ and segment $C D$, respectively, such that points $P, M$, and $Q$ are collinear. Suppose that $m \angle A B C=160^{\circ}$ and $m \angle Q M C=40^{\circ}$. Find $M P \cdot M Q$, given that $M C=6$.

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task_id
b72ffa97-c933-50bb-b446-b1344ea88212
task_key
test--b72ffa97-c933-50bb-b446-b1344ea88212
task_revision_id
1
config / split
/ test

Official source

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