1969 Canadian MO Problems/Problem 4
Problem
Let be an equilateral triangle, and
be an arbitrary point within the triangle. Perpendiculars
are drawn to the three sides of the triangle. Show that, no matter where
is chosen,
.
Solution
Let a side of the triangle be and let
denote the area of
Note that because
Dividing both sides by
, the sum of the perpendiculars from
equals
(It is independant of point
) Because the sum of the sides is
, the ratio is always