$D,E,F$ are mid points of sides $BC,CA,AB$ of triangle $ABC$ and the circumcircles of DEF,ABC touch each other then find $\sum{\cos^2A}$.
The problem is that I am able to imagine a scenario where this would happen to draw the figure and start off with solving the problem. Will the two circles touch at any of the vertex or some other intermediate point?
Could someone please help me solve this problem?
Thanks!
WLOG let $\angle A$ be the largest angle in $ABC$.
It's pretty easy to show $AFE$ and $DEF$ are congruent, furthermore $AFE$ and $ABC$ are homothetic with respect to point $A$ so the circumcircles of $AFE$ and $ABC$ touches at point $A$.
From this point it's pretty easy to see the circumcircle of $DEF$ will intersect the circumcircle of $ABC$ $0$ times when $\angle A < 90^{\circ}$, and $2$ times when $\angle A > 90^{\circ}$. When $\angle A = 90^{\circ}$ the circumcircles of $AFE$ and $DEF$ coincides and we are good.
Then the sum of cosines is just $0+1=1$