1970 IMO Problems/Problem 5
In the tetrahedron , angle is a right angle. Suppose that the foot of the perpendicular from to the plane in the tetrahedron is the intersection of the altitudes of . Prove that
For what tetrahedra does equality hold?
Let us show first that angles and are also right. Let be the intersection of the altitudes of and let meet at . Planes and are perpendicular and is perpendicular to the line of intersection . Hence is perpendicular to the plane and hence to . So Also Therefore But so , so angle . But angle , so is perpendicular to the plane , and hence angle = . Similarly, angle . Hence . But now we are done, because Cauchy's inequality gives We have equality if and only if we have equality in Cauchy's inequality, which means
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