Ratio of Areas in a Square

Given: a square with side length and diagonal . Place point on and let . Connect to divide the square into four areas. Given , calculate the ratio of areas. For example, let .

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The key here is that the height of equals the height of , so the ratio of the areas between those two triangles is .
Hence, .
Similarly, comparing areas, . Therefore |.
Substituting gives .
At this point, one can calculate the height of as .
Then and .
Also, .
After simplification, the ratio of the four areas is: .

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