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be a convex quadrilateral with 
, 
, 
and 
. Suppose![\[a^2+b^2+c^2+d^2=ab+bc+cd+da,\]](http://latex.artofproblemsolving.com/b/9/1/b91923ef3e79ecb01ed9aa8ebe3f4b3db852ea41.png)
and the area of is 
sq. units. If the length of one of the diagonals is 
units, determine the length of the other diagonal.
be a point inside the triangle 
. 
intersects 
at 
. Points 
and 
are defined similarly. Line 
intersects 
at points 
, 
. The tangents at these points to intersect BC at points 
, 
. Points 
, 
, 
, 
are defined similarly. Prove that points , , , , , lie on one conic
people and 
colors, where each person has one ball of each color. For each person, their balls have positive mass summing to one. Find the smallest constant 
such that regardless of the mass distribution, each person can choose one ball such that the sum of the chosen balls of each color does not exceed .
be the center of the circle of the triangle with 
. Furthermore, let 
and 
be points on the rays 
and 
, such that 
and 
. Show that 
bisects the segment .
