Solution for [1] Can you solve this geometry problem (Areas) | Crossed Ladder Theorem

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.