Difference between revisions of "2020 OIM Problems/Problem 1"
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== Problem == | == Problem == | ||
| − | Let ABC be an acute triangle such that AB < AC. The midpoints of | + | Let <math>ABC</math> be an acute triangle such that <math>AB < AC</math>. The midpoints of sides <math>AB</math> and <math>AC</math> are <math>M</math> and <math>N</math>, respectively. Let <math>P</math> and <math>Q</math> be points on the line <math>MN</math> such that <math>\angle CBP = \angle ACB</math> and <math>\angle QCB = \angle CBA</math>. The circumcircle of triangle <math>ABP</math> intersects the line <math>AC</math> in <math>D</math> (<math>D \ne A</math>) and the circumcircle of the triangle <math>AQC</math> intersects the line <math>AB</math> in <math>E</math> (<math>E \ne A</math>). Show that the lines <math>BC</math>, <math>DP</math> and <math>EQ</math> are concurrent. |
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| − | \CBP = \ACB and \QCB = \CBA. The circumcircle of triangle ABP intersects | ||
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| − | line AB in E (E | ||
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com | ~translated into English by Tomas Diaz. ~orders@tomasdiaz.com | ||
Latest revision as of 12:43, 14 December 2023
Problem
Let 
be an acute triangle such that 
. The midpoints of sides 
and 
are 
and 
, respectively. Let 
and 
be points on the line 
such that 
and 
. The circumcircle of triangle 
intersects the line in 
(
) and the circumcircle of the triangle 
intersects the line in 
(
). Show that the lines 
, 
and 
are concurrent.
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
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