1994 IMO Problems/Problem 2
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Let be an isosceles triangle with
.
is the midpoint of
and
is the point on the line
such that
is perpendicular to
.
is an arbitrary point on
different from
and
.
lies on the line
and
lies on the line
such that
are distinct and collinear. Prove that
is perpendicular to
if and only if
.
Solution
Let and
be on
and
respectively such that
. Then, by the first part of the problem,
. Hence,
is the midpoint of
and
, which means that
is a parallelogram. Unless
and
, this is a contradiction since
and
meet at
. Therefore,
and
, so
, as desired.