2020 CMC 12B Problems/Problem 19

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Let $ABCD$ be a convex quadrilateral such that $AB=4, BC=4, CD=3, DA=7$. There exists a unique point $P$ inside quadrilateral $ABCD$ such that the areas of $\triangle PAB, \triangle PBC, \triangle PCD, \triangle PDA$ are all numerically equal. What is the value of $PA^2+PB^2+PC^2+PD^2$?

Solution

suppose $A, B, C$ are collinear then quadrilateral $ABCD$ becomes a triangle with sides $3, 7, 8$

See also

2020 CMC 12 (ProblemsAnswer KeyResources)
Preceded by
Problem 18
Followed by
Problem 20
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All CMC 12 Problems and Solutions

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