and 
.
and 
.
Equality holds when 
or 
be a cyclic quadrilateral whose opposite sides are not parallel. Suppose points 
lie in the interiors of segments 
respectively, such that 
Let 
intersect 
at 
,
intersect 
at 
.
and 
are either parallel or coincide.
.
Where 
Y by
Let 
be a triangle with 
as midpoints of the segments 
respectively. Suppose that 
is the intersection of angle bisectors of 
and 
is the intersection of angle bisectors of 
. Denote 
as the circle of center and tangent to 
at 
, 
as the circle of center and tangent to 
at 
.
a) Prove that
is parallel to 
.
b) Prove that the radical axis of two circles
bisects the segment .
be a triangle with 
as midpoints of the segments 
respectively. Suppose that 
is the intersection of angle bisectors of 
and 
is the intersection of angle bisectors of 
. Denote 
as the circle of center and tangent to 
at 
, 
as the circle of center and tangent to 
at 
.a) Prove that
is parallel to 
.b) Prove that the radical axis of two circles
bisects the segment .
by angle chasing. If we let 
then 
due to 
being the incenter of 
.
,
Additionally, we see 
implying 
,
.
and 
.
hence 
.
is the midpoint of 
which implies ![[asy]
import graph; size(16cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = 1.6521126872149106, xmax = 27.264956864035277, ymin = -7.498208401550829, ymax = 9.547273263162316; /* image dimensions */
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/* draw figures */
draw((4.44,-1.12)--(7.6,7.02), linewidth(1));
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draw((17.76,-0.94)--(4.44,-1.12), linewidth(1) + blue);
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draw((xmin, 0.694470665794304*xmin-5.765888042271775)--(xmax, 0.694470665794304*xmax-5.765888042271775), linewidth(1) + dotted); /* line */
draw((xmin, -3.6421405488184293*xmin + 49.22234215901768)--(xmax, -3.6421405488184293*xmax + 49.22234215901768), linewidth(1) + dotted); /* line */
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draw((4.44,-1.12)--(3.6095689751522393,-3.2591482728673355), linewidth(1));
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draw((11.903329684163333,1.0393365914840254)--(14.855567355487052,1.079231695150561), linewidth(1) + dotted + wwzzqq);
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dot((4.44,-1.12),dotstyle);
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dot((7.6,7.02),dotstyle);
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dot((17.76,-0.94),dotstyle);
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label("$E$", (11.379638872836583,1.3591950289349277), NE * labelscalefactor);
label("$\omega_2$", (13.566101180613932,1.2030191498079748), NE * labelscalefactor,ffqqff);
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dot((5.564904863138467,0.9536822020107177),linewidth(4pt) + dotstyle);
label("$S_1$", (5.288779586885398,0.042855476293467705), NE * labelscalefactor);
dot((14.855567355487052,1.079231695150561),linewidth(4pt) + dotstyle);
label("$S_2$", (15.038616612382352,0.19903135542042058), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]](http://latex.artofproblemsolving.com/0/1/6/01697dc72c1c357de463f73133dfab414ce0a13a.png)
