and 
,
.
and 
and 
are on the same side of 
be midpoint of segment 
be intersection of 
with 
and 
that verify the following equation for all 
:
be positive real integer such that 
Prove that
such that 
and 
that satisfy the equation 
.
be a circle, and let a line 
intersect 
Circles 
and 
are internally tangent to 
respectively, and both are tangent to 
Similarly, circles 
and 
are externally tangent to 
respectively.
the segment 
has a length of 
and 
find the length of segment 
and 
Prove that
.
Prove that
and 
Prove that
,
,
is not greater than 
.
Y by Adventure10 and 1 other user
Let 
be an acute triangle with altitudes 
, 
, 
where 
, 
, 
lie on 
, 
, 
, respectively. Let 
be the midpoint of . The circumcircle of triangle 
cuts the line 
at 
and 
. The line cuts the line at 
. Let 
be the point of intersection of and 
. Show that the lines 
and are parallel.
be an acute triangle with altitudes 
, 
, 
where 
, 
, 
lie on 
, 
, 
, respectively. Let 
be the midpoint of . The circumcircle of triangle 
cuts the line 
at 
and 
. The line cuts the line at 
. Let 
be the point of intersection of and 
. Show that the lines 
and are parallel.
is cyclic, hence the two lines are parallel.
are both cyclic, so 
,
is cyclic, hence 
,
is cyclic, and we have 
,
.
,![\[{YZ \parallel BC \iff Y(Z,M;B,C) = -1 \iff B(Z, A; YB\cap AD, H) = - 1 \iff (X, A; Y, P) = -1,}\]](http://latex.artofproblemsolving.com/b/4/d/b4d272344235e1c1f808e3bea52e44c5a0b98e84.png)
which is true iff 
of 
But the polar of 
But also 
,
we get 
,
,
.
,
is parallel to 
where we use the fact that 
is tangent to 
:
is tangent to the 
.
;
and 
and 
are conclic and so 
and hence, 
is conclic.
and 
and so 
and 
and 
is conclic and so 
.
![[asy]
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.
and 
.
we have this because of the three tangents lemma.
is a cyclic quad.
and we have that 
(look at the triangle 
)
.
and 
are tangent to 
cyclic, and this is true iff 
,
,
,
iff the tangency is true).
finishes the problem.
.
.
