1970 IMO Problems/Problem 5

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Problem

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?

Solution

Let us show first that angles ADB and ADC are also right. Let H be the intersection of the altitudes of ABC and let CH meet AB at X. Planes CED and ABC are perpendicular and AB is perpendicular to the line of intersection CE. Hence AB is perpendicular to the plane CDE and hence to ED. So BD^2 = DE^2 + BE^2. Also CB^2 = CE^2 + BE^2. Therefore CB^2 - BD^2 = CE^2 - DE^2. But CB^2 - BD^2 = CD^2, so CE^2 = CD^2 + DE^2, so angle CDE = 90°. But angle CDB = 90°, so CD is perpendicular to the plane DAB, and hence angle CDA = 90°. Similarly, angle ADB = 90°. Hence AB^2 + BC^2 + CA^2 = 2(DA^2 + DB^2 + DC^2). But now we are done, because Cauchy's inequality gives (AB + BC + CA)^2 = 3(AB^2 + BC^2 + CA^2). We have equality if and only if we have equality in Cauchy's inequality, which means AB = BC = CA.