Consider a right triangle with legs of length h and t.






Select a fraction k between 0 and 1 and mark a length on the hypotenuse that is k times the distance from the end of the segment of length t.





Construct a perpendicular from this point to the segment of length t to construct a similar triangle with sides of length ka, kh, and kt.




Another SIMILAR triangle can be constructed by drawing a perpendicular to the segment of length h from the upper vertex of our smaller triangle.





The sides of this new triangle will have lengths of (1-k)a, (1-k)t, and (1-k)h. This is one side of the picture for our original triangle and the inscribed rectangle, completed as follows:


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