Question 43
Let $ABCD$ be a quadrilateral whose diagonals are perpendicular and intersect at $P$. Let $h_{1}, h_{2}, h_{3}, h_{4}$ be the lengths of the altitudes from $P$ to $AB, BC, CD, DA$. Show that
$$
\frac{1}{h_{1}^{2}}+\frac{1}{h_{3}^{2}}=\frac{1}{h_{2}^{2}}+\frac{1}{h_{4}^{2}}
$$
$$
\frac{1}{h_{1}^{2}}+\frac{1}{h_{3}^{2}}=\frac{1}{h_{2}^{2}}+\frac{1}{h_{4}^{2}}
$$