The red and green child-circles squeeze their father (the blue circle) in the back seat while their mother drives. Show that the orange points where the children sit and where the children are squeezing their father are concyclic.

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## Solution

Orange angle B = (1/2) (A+C), and orange angle G = (1/2)(E+F). Thus B+G = (1/2)(A+C+E+F) = (1/2) 360 = 180 degrees. Thus the orange quadrilateral is cyclic.

Thus, this proves the case when the car’s back seat is a circle instead of a line, and when the father has been lifted off the seat by the children squeezing inward.