Difference between revisions of "2017 OIM Problems/Problem 2"
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== Problem == | == Problem == | ||
| − | Let <math>ABC</math> be a right triangle and <math>\Gamma</math> its circumcircle. Let <math>D</math> be a point on the segment <math>BC</math>, distinct from <math>B</math> and | + | Let <math>ABC</math> be a right triangle and <math>\Gamma</math> its circumcircle. Let <math>D</math> be a point on the segment <math>BC</math>, distinct from <math>B</math> and <math>C</math>, and let <math>M</math> be the midpoint of <math>AD</math>. The line perpendicular to <math>AB</math> passing through <math>D</math> cuts <math>AB</math> at <math>E</math> and <math>\Gamma</math> at <math>F</math>, with point <math>D</math> between <math>E</math> and <math>F</math>. The lines <math>FC</math> and <math>EM</math> intersect at the point <math>X</math>. If <math>\angle DAE = \angle AFE</math>, show that the line <math>AX</math> is tangent to <math>\Gamma</math>. |
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com | ~translated into English by Tomas Diaz. ~orders@tomasdiaz.com | ||
Latest revision as of 13:38, 14 December 2023
Problem
Let 
be a right triangle and 
its circumcircle. Let 
be a point on the segment 
, distinct from 
and 
, and let 
be the midpoint of 
. The line perpendicular to 
passing through cuts at 
and at 
, with point between and . The lines 
and 
intersect at the point 
. If 
, show that the line 
is tangent to .
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
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