Circumcircles meeting at a common point

• Apr 10th 2009, 06:04 AM
X1337
Circumcircles 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 AM
red_dog
Let M be the point of intersection of the circumcircles of the triangles BDE and EFC. Then,

$\widehat{DME}=180-B$

$\widehat{EMF}=180-C$

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

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