is given where 
are positive reals. Prove that:
be a triangle and let 
be a circle in its plane passing through 
and 
Suppose there exist circles 
such that for 
is externally tangent to 
and passes through 
and 
where 
for all 
.
and 
have common external tangents 
and 
meeting at 
.
and 
.
through 
is convex.
and 
meet at point 
,
and 
meet at point 
.
Y by Mango247
Let 
be a triangle with 
. Suppose that the tangent line at 
to the circle passing through 
intersects the line 
at 
. Let 
be the midpoint of 
and let 
be a point on 
such that 
is parallel to .
Prove that the lines and 
are perpendicular.
be a triangle with 
. Suppose that the tangent line at 
to the circle passing through 
intersects the line 
at 
. Let 
be the midpoint of 
and let 
be a point on 
such that 
is parallel to .Prove that the lines
and 
are perpendicular.Attachments:
This post has been edited 6 times. Last edited by mikestro, Jul 4, 2020, 4:13 AM
![[asy]
import geometry;
import olympiad;
size(250);
pair a, b, c, d, e, f;
a = dir(0);
b = dir(180);
c = dir(125);
d = (sec(125*pi/180),0);
e = (d+c)/2;
f = extension(e, b, a, a+d-c);
draw(unitcircle);
draw(a--b--c--a);
draw(b--d--c);
draw(e--f--a);
draw(c--f, dashed);
dot("$A$",a,dir(a));
dot("$B$",b,dir(225));
dot("$C$",c,dir(c));
dot("$D$",d,dir(d));
dot("$E$",e,dir(e));
dot("$F$",f,dir(f));
[/asy]](http://latex.artofproblemsolving.com/6/e/6/6e645d8e457d47f2c5daa69d8f659e07a62730a5.png)
.
,
.
.
and
The intersection of these lines is 
then the 
-
.
,
is equal.
which is obvious.
so 
and since 
we get 
as desired.
. Suppose that the tangent line at 
intersects the line AB at 
and 
,
QED
