How to construct a perpendicular from hypotenuse to vertex containing 90° angle with compass
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Here's a construction, where I'll use ss for the given hypotenuse:
Construct right △OAB△OAB with |OA¯¯¯¯¯¯¯¯|=|OB¯¯¯¯¯¯¯¯|=s|OA¯|=|OB¯|=s.
Construct AC¯¯¯¯¯¯¯¯AC¯ perpendicular to AB¯¯¯¯¯¯¯¯AB¯, with |AC¯¯¯¯¯¯¯¯|=r/2|AC¯|=r/2.
Construct DD where BD←→BD↔ meets the "far side" the the circle about CCthrough
Construct right △OAB△OAB with |OA¯¯¯¯¯¯¯¯|=|OB¯¯¯¯¯¯¯¯|=s|OA¯|=|OB¯|=s.
Construct AC¯¯¯¯¯¯¯¯AC¯ perpendicular to AB¯¯¯¯¯¯¯¯AB¯, with |AC¯¯¯¯¯¯¯¯|=r/2|AC¯|=r/2.
Construct DD where BD←→BD↔ meets the "far side" the the circle about CCthrough
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