Problem

The angle made by perpendicular lines drawn from two vertices of an acute triangle to their opposite sides is equal to the supplementary angle of the remaining vertex.

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Solution

Firstly,

$$∠BOC=∠DOE.$$

Next, regarding the quadrangle $ADOE$,

$$∠A+∠DOE=2∠R,$$

$$∴ \ ∠A+∠BOC=2∠R.$$

Therefore, $∠BOC$ is equal to the supplementary angle to $∠A$.

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Reference

Teiichiro Sasabe (1976) The Encyclopedia of Geometry (2^nd edition), Seikyo-Shinsha, p.16