1970 IMO Problems/Problem 5

Problem

In the tetrahedron $ABCD$ , angle $BDC$ is a right angle. Suppose that the foot $H$ of the perpendicular from $D$ to the plane $ABC$ in the tetrahedron is the intersection of the altitudes of $\triangle ABC$ . Prove that

$( AB+BC+CA )^2 \leq 6( AD^2 + BD^2 + CD^2 )$ .

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 $E$ . 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 $CDE = 90^{\circ}$ . But angle $CDB = 90^{\circ}$ , so $CD$ is perpendicular to the plane $DAB$ , and hence angle $CDA$ = $90^{\circ}$ . Similarly, angle $ADB = 90^{\circ}$ . Hence $AB^2 + BC^2 + CA^2 = 2(DA^2 + DB^2 + DC^2)$ . But now we are done, because Cauchy's inequality gives $(AB + BC + CA)^2 \le 3(AB^2 + BC^2 + CA^2)$ (applied to vectors $(AB, BC, CA)$ and $(1, 1, 1))$ . We have equality if and only if we have equality in Cauchy's inequality, which means $AB = BC = CA.$

1970 IMO (Problems) • Resources
Preceded by Problem 4	1 • 2 • 3 • 4 • 5 • 6	Followed by Problem 6
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1970 IMO Problems/Problem 5

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