AB \cdot \sin \angle {ABC} \cdot AH + AB \cdot \sin \angle {BAC} \cdot BH + AC \cdot \sin \angle {BAC} \cdot CH = \frac{1}{2}\left(\frac{CB^4}{CC'^2} \cdot \sin^2 \angle {ABC}+{AC^4 \over AA'^2} \cdot \sin^2 \angle {ACB}+\frac{AB^4}{AA'^2} \cdot sin^2 \angle {ABC}\right)