1980 USAMO Problems/Problem 4
Problem
The insphere of a tetrahedron touches each face at its centroid. Show that the tetrahedron is regular.
Solution
Let be the tetrahedron, and let
and
be the points at which the insphere touches faces
and
respectively (and therefore the centroids of those faces). Looking at the plane containing
,
, and
, we see that the intersection of the sphere and the plane is a circle, and that
and
are both tangent to said circle.
and
are tangents to the circle from the same point and thus have the same length. The same goes for
and
. Thus, triangles
and
are congruent, and
.
Let the intersection of and
be
, and let the intersection of
and
be
. Then
and
are medians of
and
, and thus
. We already know from the previous congruence that
, and
is equal to itself. Thus,
and
are also congruent to each other. Finally,
(Because
and
are midpoints of
and
respectively), and from the congruence of
and
we have
, and again
is equal to itself. Thus
and
are congruent, thus
and
.
By applying the same logic to faces and
we get
and
. Finally, applying the same logic to faces
and
we get
and
. Putting all these equalities together, we get that all edges of the tetrahedron are equal, and thus the tetrahedron is regular.
See Also
1980 USAMO (Problems • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.