[/quote]
,
real numbers 
,
and transform the sequence into 
.
where 
are lengths of parallel sides and 
are other two sides.
?
be a cyclic quadrilateral. Let 
,
,
be the feet of the perpendiculars from 
to the lines 
,
,
,
if and only if the bisectors of 
and 
are concurrent with 
.
be reals such that 
. Prove that
.
be the feet of the altitudes from 
to 
to 
and 
as the reflections of 
be the circumcircle of 
.
with 
with 
is the circumcenter of 
can be inscribed within a circle.
and 
Prove that 
be a triangle. A circle passing through 
intersect at 
,
and 
intersect at 
.
if and only if 
.
and 
.
be reals such that 
Prove that
Prove that
such that 
.
of positive integers satisfying the following two conditions:
,
is in 
,
and 
;
such that 
is not divisible by 
.
Y by centslordm, HWenslawski, fastandfuriuos
Let 
and 
be two concentric circles with inside . Let 
and 
be two points on that are not diametrically opposite. Extend the segment 
past until it meets the circle in 
. The tangent to at and the tangent to at meet in a point 
. Draw from X the second tangent to which meets at the point 
. Show that 
bisects angle 
.
and 
be two concentric circles with inside . Let 
and 
be two points on that are not diametrically opposite. Extend the segment 
past until it meets the circle in 
. The tangent to at and the tangent to at meet in a point 
. Draw from X the second tangent to which meets at the point 
. Show that 
bisects angle 
.
Thus all the 
points 
lie on a circle with diameter 
.
are concyclic then angle chase with equal arcs
)
![[asy]
size(7cm);
import geometry;
point O=(-3,1);
point P_1=(3,1);
point Q_2=(-8,7);
circle C_1=circle(O,length(segment(O,P_1)));
circle C_2=circle(O,length(segment(O,Q_2)));
point P_2=intersectionpoints(C_1,P_1--Q_2)[1];
circle omega=circle(O,Q_2,P_1);
point Q_1=intersectionpoints(C_2,omega)[0];
point X=intersectionpoints(bisector(Q_1,Q_2),omega)[1];
draw(C_1);
draw(C_2);
draw(omega,dotted);
draw(P_1--Q_2);
draw(Q_1--X--Q_2);
draw(X--O);
draw(O--P_1--Q_1);
draw(O--Q_2);
dot(O^^P_1^^Q_2^^P_2^^Q_1^^X);
label("$O$",O,SW);
label("$P_1$",P_1,SW);
label("$Q_2$",Q_2,NW);
label("$P_2$",P_2,N);
label("$Q_1$",Q_1,SE);
label("$X$",X,NE);
label("$\mathcal{C}_1$",(-7,-3),N);
label("$\mathcal{C}_2$",(-6.5,-6),N);
[/asy]](http://latex.artofproblemsolving.com/e/1/7/e1716622b35c8ddfca96843c6688146b51809e5e.png)
and 
Since 
we know 
is cyclic. Hence, 
stands for small arc 
,
is cyclic (with diameter 
as tangents of circle 
lie on a circle
is a cyclic quad 
