For four points to be concyclic, they must be the vertices of a cyclic quadrilateral. If two opposite angles of a quadrilateral sum to 180º it is cyclic. This is what I will show below.
Call the centers of the circles in order A, B, C, D. Call points of tangency W,X,Y,Z so that we can draw quadrilaterals ABCD and WXYZ.
Being a quadrilateral makes angles A+B+C+D=360
There are 4 isosceles triangles which allow us to find:
Angle AWZ = 180A/2
Angle BWC = 180B/2 etc.
Angle ZWX = 180(A+B)/2
Angle XYZ = 180(C+D)/2 etc.
Angles ZWX and XYZ are the opposite angles mentioned in the first paragraph. Their sum is
360(A+B+C+D)/2 = 360  360/2 = 180
QED

Posted by Jer
on 20130704 01:27:18 