1995 IMO Problems/Problem 1
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Problem
Let 
be four distinct points on a line, in that order. The circles with diameters 
and 
intersect at 
and 
. The line 
meets 
at 
. Let 
be a point on the line other than . The line 
intersects the circle with diameter at 
and 
, and the line 
intersects the circle with diameter at 
and 
. Prove that the lines 
are concurrent.
Solution
Since is on the circle with diameter , we have 
and so 
. We simlarly find that 
. Also, notice that the line is the radical axis of the two circles with diameters and . Thus, since is on , we have 
and so by the converse of Power of a Point, the quadrilateral 
is cyclic. Thus, 
. Thus, 
and so quadrilateral 
is cyclic. Let the circle which contains the points be cirle 
. Then, the radical axis of and the circle with diameter is line 
. Also, the radical axis of and the circle with diameter is line 
. Since the pairwise radical axes of 3 circles are concurrent, we have are concurrent as desired.
