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 (applied to vectors $(AB, BC, CA)$ and $(1, 1, 1)$ ) gives $(AB + BC + CA)^2 \le 3(AB^2 + BC^2 + CA^2)$ .

We have equality if and only if we have equality in Cauchy's inequality, which means $AB = BC = CA.$

Solution 2

Let $x = DH, a = BC, b = CA, c = AB$

The plan of this proof is to compute $HA, HB, HC$ in terms of $a, b, c$ , then compute $DA^2, DB^2, DC^2$ in terms of $a, b, c, x$ , impose the condition that $\angle BDC = \pi/2$ to determine $x$ , and calculate $DA^2 + DB^2 + DC^2$ in terms of $a, b, c$ . The problem will become a simple inequality in $a, b, c$ which will be easy to prove.

Formulas for the distance from the orthocenter to the vertices are reasonably well known, but to make this solution self contained, we compute them here. From $\triangle ABE_B$ we have that $BE_B = c \sin A$ . From $\triangle CBE$ we have $BE = a \cos B$ . From $\triangle ABE_B \sim \triangle HBE$ we have $\frac{HB}{c} = \frac{BE}{BE_B}$ . It follows that $HB = c \ \frac{a \cos B}{c \sin A} = \frac{a \cos B}{\sin A}$ .

Similarly, we have $HC = \frac{a \cos C}{\sin A}$ and $HA = \frac{b \cos A}{\sin B} = \frac{a \cos A}{\sin A}$ .

(In the last equality we used the Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}.$ We want to have $\sin A$ at the denominator to simplify computations.)

Then $DA^2 = x^2 + \frac{a^2 \cos^2 A}{\sin^2 A}, DB^2 = x^2 + \frac{a^2 \cos^2 B}{\sin^2 A}, DC^2 = x^2 + \frac{a^2 \cos^2 C}{\sin^2 A}$ .

Since $\angle BDC = \pi/2, DB^2 + DC^2 = a^2$ . This yields $x^2 = \frac{a^2}{2 \sin^2 A} \ (\sin^2 A - \cos^2 B - \cos^2 C)$ .

Using this value for $x^2$ we get $DA^2 + DB^2 + DC^2 = \frac{a^2}{2 \sin^2 A} \ (3 \sin^2 A + 2 \cos^2 A - \cos^2 B - \cos^2 C)$

After some simplifications, and using again the Law of Sines, this becomes

$DA^2 + DB^2 + DC^2 = \frac{a^2}{2 \sin^2 A} \ (\sin^2 A + \sin^2 B + \sin^2 C) = \frac{a^2}{2} + \frac{a^2}{2 \sin^2 A} \sin^2 B + \frac{a^2}{2 \sin^2 A} \sin^2 C =$

$\frac{a^2}{2} + \frac{b^2}{2 \sin^2 B} \sin^2 B + \frac{c^2}{2 \sin^2 C} \sin^2 C = \frac{1}{2} \ (a^2 + b^2 + c^2)$ .

What we need to prove then is that $(a + b + c)^2 \le 3(a^2 + b^2 + c^2)$ . We could invoke the Cauchy-Schwarz Inequality like in the first solution, but we can just as well make the computations, and notice that this is equivalent to

$2 a^2 + 2 b^2 + 2 c^2 -2ab - 2bc - 2ca \ge 0$ or $(a - b)^2 + (b - c)^2 + (c - a)^2 \ge 0$ , which is always true.

We also see that we have equality if and only if $a = b = c$ .

[Solution by pf02, December 2024]

1970 IMO (Problems) • Resources
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1970 IMO Problems/Problem 5

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Solution 2