1980 USAMO Problems/Problem 4
The insphere of a tetrahedron touches each face at its centroid. Show that the tetrahedron is regular.
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.
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