2006 Romanian NMO Problems/Grade 7/Problem 3
Problem
In the acute-angle triangle we have
. The points
and
are the feet of the altitudes from
and
, and
is the orthocenter of the triangle. We consider the points
and
on the segments
and
such that
. Prove that
a) ;
b) .
Solution
a) Note that quadrilateral is cyclic, because
. Thus
and
. Similarly
. Therefore
and
. However
is a
triangle, so
and
. By Pythagorean theorem,
. However
, so
, and thus
, or
.
b) ,
,
.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.