[/quote]
,
be a quadrilateral inscribed in a circle of center 
.
and 
,
be the intersection point of the diagonals 
are distinct. Prove that the points 
are concyclic if and only if the points 
are concyclic.
such that there exists a pair 
of positive integers, such that 
is not divisible by the cube of any prime, and 
such that 
?
be triangle with 
and 
on the segment 
such that 
. Consider point 
on 
such that 
. Point 
is the intersection of tangent line through point 
respectively . If the tangent through point 
intersect 
at 
. Line 
intersect 
at 
bisect 
.
be a triangle and 
an interior point of 
is less than or equal to 
.
.
be the midpoint of the arc 
not containing 
similarly. Show that the orthocenter of 
is the incenter 
of 
intersect 
at 
.
.
lies on the line 
be the reflection of point 
and 
be the circumcenter and the orthocenter of triangle 
.
be the points where the reflection of line 
with respect to line 
intersects the circumcircle of triangle 
is a point on the line 
such that 
,
.
passes through 
such that 
and 
at 
at 
is parallel to 
and line 
meets 
.
at point 
on the circumcircle of 
is such that 
.
and 
meet on 
be an inscribed circle of 
,
and 
be diameters of 
be the pole of 
and 
with respect to 
cuts 
cuts 
at 
at 
.
and 
be midpoint of 
and 
,
,
and perpendicular bisector of 
are concurrent.
is a parallelogram. Let 
such that 
.
at 
and 
.
.
be a non-cyclic quadrilateral. Let 
and 
be the circumcentre and the circumradius of the triangle 
.
and 
in a similar way. Prove that![\[\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.\]](http://latex.artofproblemsolving.com/f/f/5/ff575c3012e0b86c5aeeec32aae5268a4dad74d2.png)
Y by Rounak_iitr
A trapezoid 
with right angles at 
and 
has an inscribed circle with center 
and radius 
. Let the lengths of the parallel sides 
and 
be 
and 
, and the intersection of the diagonals 
and 
be 
.
1. Prove that the perpendicular from to one of the trapezoid sides has the length .
2. Determine the distance between and as a function of and .
with right angles at 
and 
has an inscribed circle with center 
and radius 
. Let the lengths of the parallel sides 
and 
be 
and 
, and the intersection of the diagonals 
and 
be 
.1. Prove that the perpendicular from
to one of the trapezoid sides has the length .2. Determine the distance between
and as a function of and .
.
of the incircle; the line 
is a tangent line if 
.
intersects the line 
in the point 
.
.
