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Another triangle in a square problem (Posted on 2023-03-29) Difficulty: 3 of 5
ABCD is a square with point E inside the square.
Line segments AE, BE, and DE are drawn.
The length of AE is 8 and angle AEB is a right angle.

What is the area of triangle AED?

See The Solution Submitted by Brian Smith    
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Solution my solution Comment 4 of 4 |
I solved it by assuming that there was a unique answer:

If this is solvable (we have been given enough information) then WLOG we can assume E is at the center of the square so that all four angles at the center between A and B, B and C, C and D, and D and E are right angles. The triangle in question is a right triangle with legs equal to 8, and has an area of 8^2 / 2  = 32.

The same is true when E is coincident with B, which is of course on the semicircle connecting A with B that forms the locus of E when AEB is a right angle.

When is or is approaching A, this can only be a limit as E approaches A, with the requirement for a larger and larger square with a longer and longer but narrower and narrower triangle. I haven't done the calculus, but the assurance that the answer doesn't change, combined with the one or two solutions, gives confidence 32 is correct.

  Posted by Charlie on 2023-03-29 21:52:20
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