1982 USAMO Problems/Problem 5

Problem

$A,B$, and $C$ are three interior points of a sphere $S$ such that $AB$ and $AC$ are perpendicular to the diameter of $S$ through $A$, and so that two spheres can be constructed through $A$, $B$, and $C$ which are both tangent to $S$. Prove that the sum of their radii is equal to the radius of $S$.

Solution

Let the two tangent spheres be $S_1$ and $S_2$, and let $O, O_1, O_2$ and $R, R_1, R_2$ be the origins and radii of $S, S_1, S_2$ respectively. Then $AO$ stands normal to the plane $P$ through $\Delta ABC$. Because both spheres go through $A$, $B$, and $C$, the line $O_1 O_2$ also stands normal to $P$, meaning $AO$ and $O_1 O_2$ are both coplanar and parallel. Therefore the problem can be flattened to the plane $P'$ through $A$, $O$, $O_1$ and $O_2$.

Let $X, Y, Z, M, N$ be points on $P'$ such that $X = S \cap O O_1, \quad Y = S_1 \cap S_2 \neq A, \quad Z = S \cap O O_2, \quad M = O_1 Y \cap AO, \quad N = O_2 Y \cap AO$

Let $J$ be the three circles radical center, meaning $JX$ and $JZ$ are tangent segments to $S$ and $J \in AY$.

Because $Y \in P \iff \angle NAY = \angle YAM = 90 ^{\circ},$ we have that $\overline{YN}$ and $\overline{YM}$ are diameters.

This means that $\angle OZN = \angle ZNY = \angle JZY$ and $\angle MXO = \angle YMX= \angle YXJ$.

And because $\angle ZNO = 180^{\circ} - \angle AYZ = \angle ZYJ$ and $\angle OMX = 180^{\circ} -\angle XYA = \angle JYX,$ we have that $\Delta JZY \sim \Delta OZN$ and $\Delta JYX \sim \Delta OMX$.

We then conclude that $\overline{OM} = \overline{OX} \enspace \frac {\overline{JY} }{ \overline{JX}} = \overline{OZ} \enspace \frac {\overline{JY} }{ \overline{JZ}} = \overline{ON}$

Let $a \top b$ denote line $a$ bisecting line segment $b$.

Since $O_1 O_2 || MN$ it follows that $OY \, \top \, \overline{MN} \Rightarrow OY \, \top \, \overline{O_1 O_2}$. Similarly we have that $O_1 O_2 \, \top \, \overline{YM} \Rightarrow O_1 O_2 \, \top \, \overline{OY}$.

And so $O_1 O O_2 Y$ is a parallelogram because $\overline{OY}$ and $\overline{O_1 O_2}$ bisect each other, meaning $R = \overline{OZ} = \overline{O O_2} + \overline{O_2 Z} = \overline{O_1 Y}+ R_2 = R_1 + R_2$

$Quod \enspace Erat \enspace Demonstrandum$