1981 IMO Problems/Problem 1
Revision as of 19:53, 28 October 2006 by Boy Soprano II (talk | contribs)
Problem
is a point inside a given triangle
.
are the feet of the perpendiculars from
to the lines
, respectively. Find all
for which
is least.
Solution
We note that is twice the triangle's area, i.e., constant. By the Cauchy-Schwarz Inequality,
,
with equality exactly when , which occurs when
is the triangles incenter, Q.E.D.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.