BG-MO 1971 Problem 6

In a triangular pyramid $SABC$ one of the plane angles with vertex $S$ is a right angle and the orthogonal projection of $S$ on the base plane $ABC$ coincides with the orthocenter of the triangle $ABC$ . Let $SA=m$ , $SB=n$ , $SC=p$ , $r$ is the inradius of $ABC$ . $H$ is the height of the pyramid and $r_1,r_2,r_3$ are radii of the incircles of the intersections of the pyramid with the plane passing through $SA,SB,SC$ and the height of the pyramid. Prove that

(a) $m^2+n^2+p^2\ge18r^2$ ; (b) $\frac{r_1}H,\frac{r_2}H,\frac{r_3}H$ are in the range $(0.4,0.5)$ .