AoPS Academy
with 
. Denote the orthocenter of 
as 
, the intersection of 
and 
as 
. Compute the length of 
.
with 
and 
. Let 
be a point on 
such that 
. Suppose that the angle bisector of 
is tangent to the circle with diameter 
and say it intersects at point 
. Find the length of 
.
be a point such that 
is a parallelogram. Since, 
and 
, it follows that 
is also a parallelogram. Also, we have 
, since 
and 
. So, 
and this implies 
is cyclic. It is easy to see that 
. Hence, 
as required. 
![[asy]
import cse5;
pair A=(-4.42,2.33), B=(0.6,2.67), C=(-0.34,-2.21), D=(-5.36,-2.55), O=(-2,-0.01), O1=(-1.82,5.01);
D(MP("A",D(A),NW)--MP("B",D(B),NE)--MP("C",D(C),SE)--MP("D",D(D),SW)--cycle);
D(MP("O",D(O),S)--C);
D(MP("O",D(O),S)--A);
D(MP("O",D(O),S)--B);
D(MP("O",D(O),S)--D);
D(MP("O'",D(O1),N)--A,dashed);
D(O1--B,dashed);
D(circumcircle(O,A,B),dashed);
D(O1--O,dashed);
[/asy]](http://latex.artofproblemsolving.com/7/f/2/7f2f0b02f8b4b3836295c480af2d299f54240e0d.png)
is a parallelogram and 
be a point inside this parallaogram such that 
°
.