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Difference between revisions of "2022 OIM Problems/Problem 1"

(Created page with "== Problem == Let <math>ABC</math> be an equilateral triangle with circumcenter <math>O</math> and circumcircle <math>\Gamma</math>. Let <math>D</math> be a point on the mino...")
 
 
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== Problem ==
 
== Problem ==
  
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Let <math>ABC</math> be an equilateral triangle with circumcenter <math>O</math> and circumcircle <math>\Gamma</math>. Let <math>D</math> be a point on the minor arc <math>BC</math>, with <math>DB > DC</math>. The perpendicular bisector of <math>OD</math> intersects <math>\Gamma</math> at <math>E</math> and <math>F4, with </math>E<math> on the minor arc </math>BC<math>. Let </math>P<math> be the intersection point of lines </math>BE<math> and </math>CF<math>.  Prove that </math>PD<math> is perpendicular to </math>BC$.
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Let <math>ABC</math> be an equilateral triangle with circumcenter <math>O</math> and circumcircle <math>\Gamma</math>. Let <math>D</math> be a point on the minor arc <math>BC</math>, with <math>DB > DC</math>. The perpendicular bisector of <math>OD</math> intersects <math>\Gamma</math> at <math>E</math> and <math>F</math>, with <math>E</math> on the minor arc <math>BC</math>. Let <math>P</math> be the intersection point of lines <math>BE</math> and <math>CF</math>.  Prove that <math>PD</math> is perpendicular to <math>BC</math>.
  
 
== Solution ==
 
== Solution ==

Latest revision as of 02:29, 14 December 2023

Problem

Let $ABC$ be an equilateral triangle with circumcenter $O$ and circumcircle $\Gamma$. Let $D$ be a point on the minor arc $BC$, with $DB > DC$. The perpendicular bisector of $OD$ intersects $\Gamma$ at $E$ and $F$, with $E$ on the minor arc $BC$. Let $P$ be the intersection point of lines $BE$ and $CF$. Prove that $PD$ is perpendicular to $BC$.

Solution

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See also

https://sites.google.com/uan.edu.co/oim-2022/inicio

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