The solution to the puzzle involves a variant of the Crossed Ladder Theorem, which I will call the "Extended Crossed Ladder Theorem", which states,
1/c + 1/f = 1/d + 1/e
For our puzzle, the values of e, f, and d in the diagram to the left are clear from the areas in the diagram above:
e = (5+10)/w
f = 10/w
d = (8+10)/w,
where w is (1/2)(AB).
From the Extended Crossed Ladder Theorem, w/c + w/10 = w/18 + w/15, so c=45/w, which means the area of the entire triangle is 45. To find the indicated area, we subtract the other areas from 45, so the answer to the puzzle is
45 - 10 - 5 - 8 = 22.
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