Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2005 IMO Problems/Problem 5"

(Created page with "Let <math>ABCD</math> be a fixed convex quadrilateral with <math>BC = DA</math> and <math>BC \nparallel DA</math>. Let two variable points <math>E</math> and <math>F</math> li...")
 
 
Line 1: Line 1:
 +
==Problem==
 +
 
Let <math>ABCD</math> be a fixed convex quadrilateral with <math>BC = DA</math> and <math>BC \nparallel DA</math>. Let two variable
 
Let <math>ABCD</math> be a fixed convex quadrilateral with <math>BC = DA</math> and <math>BC \nparallel DA</math>. Let two variable
−
points <math>E</math> and <math>F</math> lie of the sides <math>BC</math> and <math>DA</math>, respectively, and satisfy <math>BE = DF</math>. The lines <math>AC</math>
+
points <math>E</math> and <math>F</math> lie of the sides <math>BC</math> and <math>DA</math>, respectively, and satisfy <math>BE = DF</math>. The lines <math>AC</math> and <math>BD</math> meet at <math>P</math>, the lines <math>BD</math> and <math>EF</math> meet at <math>Q</math>, the lines <math>EF</math> and <math>AC</math> meet at <math>R</math>. Prove that the circumcircles of the triangles <math>PQR</math>, as <math>E</math> and <math>F</math> vary, have a common point other than <math>P</math>.
−
and <math>BD</math> meet at <math>P</math>, the lines <math>BD</math> and <math>EF</math> meet at <math>Q</math>, the lines <math>EF</math> and <math>AC</math> meet at <math>R</math>. Prove
+
 
−
that the circumcircles of the triangles <math>PQR</math>, as <math>E</math> and <math>F</math> vary, have a common point other than
+
==Solution==
−
<math>P</math>.
+
{{solution}}
 +
 
 +
==See Also==
 +
 
 +
{{IMO box|year=2005|num-b=4|num-a=6}}

Latest revision as of 01:00, 19 November 2023

Problem

Let $ABCD$ be a fixed convex quadrilateral with $BC = DA$ and $BC \nparallel DA$. Let two variable points $E$ and $F$ lie of the sides $BC$ and $DA$, respectively, and satisfy $BE = DF$. The lines $AC$ and $BD$ meet at $P$, the lines $BD$ and $EF$ meet at $Q$, the lines $EF$ and $AC$ meet at $R$. Prove that the circumcircles of the triangles $PQR$, as $E$ and $F$ vary, have a common point other than $P$.

Solution

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

See Also

2005 IMO (Problems) • Resources
Preceded by
Problem 4 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Problem 6
All IMO Problems and Solutions
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2005_IMO_Problems/Problem_5&oldid=204519"