PRMO 2013 question 15

15. Let \(A_1,B_1,C_1,D_1\) be the midpoints of the sides of a convex quadrilateral \(ABCD\), and let \(A_2,B_2,C_2,D_2\) be the midpoints of the sides of the quadrilateral \(A_1B_1C_1D_1\). If \(A_2B_2C_2D_2\) is a rectangle with side lengths 4 and 6, what is the product of the lengths of the diagonals of \(ABCD\)?

Answer: 208

Solution:
Assume that ABCD is a rectangle and draw all the points.
Construct the diagram and use TMT(Triangle Midsegment Theorem) and symmetry.
You will realized that diagonals of A1B1C1D1 are 8,12 and they are also the sides of ABCD.
So diagonal length of ABCD will be Sqrt(208). Hence the answer 208.

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