1992 USAMO Problems/Problem 4
Problem
Chords ,
, and
of a sphere meet at an interior point
but are not contained in the same plane. The sphere through
,
,
, and
is tangent to the sphere through
,
,
, and
. Prove that
.
Solution
Consider the plane through . This plane, of course, also contains
. We can easily find the
is isosceles because the base angles are equal. Thus,
. Similarly,
. Thus,
. By symmetry,
and
, and hence
as desired.
Resources
1992 USAMO (Problems • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |