Two overlapping rectangles. What is the proportion of their areas? All triangles in the diagram are right.
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The rectangles have equal area, so their proportion is 1.
Say the green rectangle has dimensions a and b. Draw a help line from its top left corner parallel to the long orange side. From comparing angles it is easily seen that the orange rectangle has dimensions a/cosα and bcosα. In taking its area, the cosines cancels.