Question 47
Let $ABCD$ be a parallelogram with acute angle at $A$. Let $E$ be the foot of the perpendicular from $C$ to the line $AB$, and let $F$ be the foot of the perpendicular from $C$ to the line $AD$.
Prove that
$$
|AB| \cdot |AE| + |AD| \cdot |AF| = |AC|^2. \quad (\text{USSR})
$$
Prove that
$$
|AB| \cdot |AE| + |AD| \cdot |AF| = |AC|^2. \quad (\text{USSR})
$$