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Difference between revisions of "2021 OIM Problems/Problem 2"

(Created page with "== Problem == Consider an acute triangle <math>ABC</math>, with <math>AC > AB</math>, and let <math>\Gamma</math> be its circumcircle. Let <math>E</math> and <math>F</math>...")
 
 
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== Problem ==
 
== Problem ==
  
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Consider an acute triangle <math>ABC</math>, with <math>AC > AB</math>, and let <math>\Gamma</math> be its circumcircle.  Let <math>E</math> and <math>F</math> be the midpoints of the sides <math>AC</math> and <math>AB</math>, respectively. The circumcircle of triangle <math>CEF</math> intersects <math>\Gamma</math> at <math>X4 and </math>C<math>, with </math>X \ne C<math>. The line </math>BX<math> and the line tangent to </math>\Gamma<math> at </math>A<math> intersect at </math>Y<math>. Let </math>P<math> be the point on segment </math>AB<math> such that </math>YP = YA<math>, with </math>P \ne A<math>, and let </math>Q<math> be the point where </math>AB<math> intersects the line parallel to </math>BC<math> passing through </math>Y<math>. Show that </math>F<math> is the midpoint of </math>PQ$.
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Consider an acute triangle <math>ABC</math>, with <math>AC > AB</math>, and let <math>\Gamma</math> be its circumcircle.  Let <math>E</math> and <math>F</math> be the midpoints of the sides <math>AC</math> and <math>AB</math>, respectively. The circumcircle of triangle <math>CEF</math> intersects <math>\Gamma</math> at <math>X</math> and <math>C</math>, with <math>X \ne C</math>. The line <math>BX</math> and the line tangent to <math>\Gamma</math> at <math>A</math> intersect at <math>Y</math>. Let <math>P</math> be the point on segment <math>AB</math> such that <math>YP = YA</math>, with <math>P \ne A</math>, and let <math>Q</math> be the point where <math>AB</math> intersects the line parallel to <math>BC</math> passing through <math>Y</math>. Show that <math>F</math> is the midpoint of <math>PQ</math>.
  
 
'''Note:''' The circumcircle of a triangle is the circle passing through its three vertices.
 
'''Note:''' The circumcircle of a triangle is the circle passing through its three vertices.

Latest revision as of 02:53, 14 December 2023

Problem

Consider an acute triangle $ABC$, with $AC > AB$, and let $\Gamma$ be its circumcircle. Let $E$ and $F$ be the midpoints of the sides $AC$ and $AB$, respectively. The circumcircle of triangle $CEF$ intersects $\Gamma$ at $X$ and $C$, with $X \ne C$. The line $BX$ and the line tangent to $\Gamma$ at $A$ intersect at $Y$. Let $P$ be the point on segment $AB$ such that $YP = YA$, with $P \ne A$, and let $Q$ be the point where $AB$ intersects the line parallel to $BC$ passing through $Y$. Show that $F$ is the midpoint of $PQ$.

Note: The circumcircle of a triangle is the circle passing through its three vertices.

Solution

This problem needs a solution. If you have a solution for it, please help us out by adding it.

See also

https://olcoma.ac.cr/internacional/oim-2021/examenes

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