board, such that Batman and Robin are on the same diagonal (picture). In each round, first The Joker moves to an adjacent cell (having a common side), without exiting the board. Then in the same round Batman and Robin move to an adjacent cell. The Joker wins if he reaches the fourth "target" vertex cell (marked T). Batman and Robin win if they catch The Joker i.e. at least one of them is on the same cell as The Joker.
Prove that
Where 
Prove that
Let 
Prove that
Let 
Prove that
be the foot of perpendicular from 
.
with centre 
at 
and 
respectively. Let 
,
is equidistant from 
,
.
with integer coefficients such that for any positive integer 
,
is positive and 
Y by Adventure10, Mango247
Two circles 
and 
intersect at 
and 
. The common tangent of and that is close to than meets at 
and at 
. 
extended meets at 
and 
extended meets at 
. Let 
and 
be the midpoints of 
and 
respectively.
Show that
bisects angle 
.
and 
intersect at 
and 
. The common tangent of and that is close to than meets at 
and at 
. 
extended meets at 
and 
extended meets at 
. Let 
and 
be the midpoints of 
and 
respectively.Show that
bisects angle 
.
,
,
and 
.
concyclic. Similarly, 
concyclic. Then 
.
.
and 
.
.
,
and 
are three similar triangles, and 
are their respective medians. The result follows.
