Problem
The two perpendicular lines drawn from the vertex of an isosceles triangle to the bisectors of the base angles are equal in length.
$$ $$
$$ $$
$\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$
$$ $$
$$ $$
$$ $$
$$ $$
$$ $$
$$ $$
$$ $$
$$ $$
$$ $$
$$ $$
Solution
Let $D$ and $E$ be the feet of the perpendicular lines drawn from $A$ to the bisectors of $∠B$ and $∠C$, respectively.
For $△ABD and $△ACE$,
$$AB=AC, \qquad ∠ABD=∠ACE \qquad and \qquad ∠BDA=∠CEA=∠R.$$
Therefore, from the problem $0031$,
$$△ABD≡△ACE,$$
$$∴ \ AD=AE.$$
$ $
$ $
$ $
Reference Teiichiro Sasabe (1976) The Encyclopedia of Geometry (2nd edition), Seikyo-Shinsha, p.33.