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without using the following method. I know that you can solve it using 
but I'm searching for an algebraic proof.
be an acute-angled triangle with circumcircle 
. A circle 
is internally tangent to at 
and also tangent to 
at 
. Let 
and 
intersect at 
and 
respectively. Let 
and 
be points on line such that 
is the midpoint of 
and 
is the midpoint of 
. Lines 
and 
meet at 
and intersect again at 
and 
respectively. The ray 
meets the circumcircle of triangle 
again at 
.
.
has as the midpoint of and orthocenter 
. Let be an arbitrary point on the nine point circle of . The line through perpendicular to 
intersects at . The line through perpendicular to 
intersects 
at 
. If is the midpoint of , show that 
.
be real numbers such that 
. Prove that
Where 
. E,F are located on AB such that 
. Prove that the C-excircle of triangle ABC touches the circumcircle (EFC).
is tangent to 
through 
then it is known that 
Inversion with center 
and power 
swaps circle 
and the sideline and takes C-excircle into itself. Since 
and are tangent, then by conformity 
and are tangent as well.
.
and 
at 
and 
respectively. It is well-known that 
.
,
and 
map to themselves and the circumcircle of 
maps to 
.
is tangent to the 
and the 
meets the 
is the tangency point between the