Difference between revisions of "2000 IMO Problems/Problem 1"
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Two circles <math>G_1</math> and <math>G_2</math> intersect at two points <math>M</math> and <math>N</math>. Let <math>AB</math> be the line tangent to these circles at <math>A</math> and <math>B</math>, respectively, so that <math>M</math> lies closer to <math>AB</math> than <math>N</math>. Let <math>CD</math> be the line parallel to <math>AB</math> and passing through the point <math>M</math>, with <math>C</math> on <math>G_1</math> and <math>D</math> on <math>G_2</math>. Lines <math>AC</math> and <math>BD</math> meet at <math>E</math>; lines <math>AN</math> and <math>CD</math> meet at <math>P</math>; lines <math>BN</math> and <math>CD</math> meet at <math>Q</math>. Show that <math>EP=EQ</math>. | Two circles <math>G_1</math> and <math>G_2</math> intersect at two points <math>M</math> and <math>N</math>. Let <math>AB</math> be the line tangent to these circles at <math>A</math> and <math>B</math>, respectively, so that <math>M</math> lies closer to <math>AB</math> than <math>N</math>. Let <math>CD</math> be the line parallel to <math>AB</math> and passing through the point <math>M</math>, with <math>C</math> on <math>G_1</math> and <math>D</math> on <math>G_2</math>. Lines <math>AC</math> and <math>BD</math> meet at <math>E</math>; lines <math>AN</math> and <math>CD</math> meet at <math>P</math>; lines <math>BN</math> and <math>CD</math> meet at <math>Q</math>. Show that <math>EP=EQ</math>. |
Revision as of 23:38, 10 December 2017
Problem
Two circles and
intersect at two points
and
. Let
be the line tangent to these circles at
and
, respectively, so that
lies closer to
than
. Let
be the line parallel to
and passing through the point
, with
on
and
on
. Lines
and
meet at
; lines
and
meet at
; lines
and
meet at
. Show that
.