Circles Cover a Quadrilateral

Here is a problem from one of the first issues of the Russian Kvant magazine (G. Galperin, M10, n 2, 1970):

Four circles centered at the vertices of a quadrilateral ABCD completely cover the quadrilateral. Prove that some three of the circles cover one of the triangles ABC, BCD, CDA, DAB.

(In the applet you can click inside a circle to make it disappear or reappear again.)

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It is relatively easy to prove the statement by considering several cases. I find this messy and solicit more elegant solutions, which appears to escape me. Please post at the CTK Exchange.

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