Difference between revisions of "1994 IMO Problems/Problem 2"
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==Solution== | ==Solution== | ||
Let <math> E'</math> and <math> F'</math> be on <math> AB</math> and <math> AC</math> respectively such that <math> E'F'\perp OQ</math>. Then, by the first part of the problem, <math> QE'=QF'</math>. Hence, <math> Q</math> is the midpoint of <math> EF</math> and <math> E'F'</math>, which means that <math> EE'FF'</math> is a parallelogram. Unless <math> E=E'</math> and <math> F=F'</math>, this is a contradiction since <math> EE'</math> and <math> F'F</math> meet at <math> A</math>. Therefore, <math> E=E'</math> and <math> F=F'</math>, so <math> OQ\perp EF</math>, as desired. | Let <math> E'</math> and <math> F'</math> be on <math> AB</math> and <math> AC</math> respectively such that <math> E'F'\perp OQ</math>. Then, by the first part of the problem, <math> QE'=QF'</math>. Hence, <math> Q</math> is the midpoint of <math> EF</math> and <math> E'F'</math>, which means that <math> EE'FF'</math> is a parallelogram. Unless <math> E=E'</math> and <math> F=F'</math>, this is a contradiction since <math> EE'</math> and <math> F'F</math> meet at <math> A</math>. Therefore, <math> E=E'</math> and <math> F=F'</math>, so <math> OQ\perp EF</math>, as desired. | ||
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+ | ==See Also== | ||
+ | {{IMO box|year=1994|num-b=1|num-a=3}} |
Latest revision as of 00:28, 22 November 2023
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.
See Also
1994 IMO (Problems) • Resources | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |