Question 1109797
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Draw a few examples which satisfy the given conditions.<br>
Start with P and Q being on diagonal AC so that angle DAP is 45 degrees.  Then move P and Q left and right (or vice versa) maintaining the requirements of the problem.<br>
You should see that the limiting case (the longest possible length of AP, PQ, and QC) is when P is on CD with DP:PC = 1:2 and Q is on AB with AQ:QB = 2:1.  In that case, the size of the angle is arctan(1/3).<br>
If P and Q must be INSIDE the square, then there is no answer to the question, because the angle can get as close as you want to arctan(1/3) without getting there.