Difference between revisions of "1981 IMO Problems/Problem 1"
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− | with equality exactly when <math> \displaystyle PD = PE = PF </math>, which occurs when <math> \displaystyle P </math> is the | + | with equality exactly when <math> \displaystyle PD = PE = PF </math>, which occurs when <math> \displaystyle P </math> is the triangle's incenter, Q.E.D. |
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Revision as of 21:34, 24 March 2007
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 triangle's 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.