Ratio of Areas in a Square
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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 
.
Contributed by: Abraham Gadalla (August 2011)
Open content licensed under CC BY-NC-SA
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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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