Hey i have this problem:

Points D,E and F lie on the sides AB, BC and CA, respectively of triangle ABC.

Prove that the circumcircles of triangles ADF, BED and CFE meet in a common point.

Thanks

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- Apr 10th 2009, 06:04 AMX1337Circumcircles meeting at a common point
Hey i have this problem:

Points D,E and F lie on the sides AB, BC and CA, respectively of triangle ABC.

Prove that the circumcircles of triangles ADF, BED and CFE meet in a common point.

Thanks - Apr 10th 2009, 07:20 AMred_dog
Let M be the point of intersection of the circumcircles of the triangles BDE and EFC. Then,

$\displaystyle \widehat{DME}=180-B$

$\displaystyle \widehat{EMF}=180-C$

$\displaystyle \widehat{DMF}=360-(180-B)-(180-C)=B+C=180-A$

That means M is on the circumcircle of the triangle ADF.