2005 IMO Problems/Problem 5: Difference between revisions
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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>. | 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>. | ||
==Solution (Video Solution) == | ==Solution (Video Solution, with an alternative version of the problem as an exercise) == | ||
https://youtu.be/Ceap-NC2nW8 | https://youtu.be/Ceap-NC2nW8 | ||
==See Also== | ==See Also== | ||
{{IMO box|year=2005|num-b=4|num-a=6}} | {{IMO box|year=2005|num-b=4|num-a=6}} | ||
Latest revision as of 07:23, 4 September 2024
Problem
Let 
be a fixed convex quadrilateral with 
and 
. Let two variable
points 
and 
lie of the sides 
and 
, respectively, and satisfy 
. The lines 
and 
meet at 
, the lines and 
meet at 
, the lines and meet at 
. Prove that the circumcircles of the triangles 
, as and vary, have a common point other than .
Solution (Video Solution, with an alternative version of the problem as an exercise)
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 | ||
