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