1969 Canadian MO Problems/Problem 10
Problem
Let be the right-angled isosceles triangle whose equal sides have length 1.
is a point on the hypotenuse, and the feet of the perpendiculars from
to the other sides are
and
. Consider the areas of the triangles
and
, and the area of the rectangle
. Prove that regardless of how
is chosen, the largest of these three areas is at least
.
Solution
Let Because triangles
and
both contain a right angle and a
angle, they are isosceles right triangles. Hence,
and
Now let's consider when or else one of triangles
and
will automatically have area greater than
In this case,
Therefore, one of these three figures will always have area greater than
regardless of where
is chosen.
1969 Canadian MO (Problems) | ||
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