convex quadrilaterals and circles

Question

Suppose you have an arbitrary convex quadrilateral call it $WXYZ$ and four circles with diameters $WX, XY, YZ, ZW$. How would you prove that the four circles would cover the whole quadrilateral completely?

Answer 1

Split on diagonal $\overline{WY}$ into $\triangle WXY$ and $\triangle YZW$.

Consider $\triangle WXY$. Draw altitude $XX^\prime$ to $WY$. Then you have right triangles $\triangle WXX^\prime$ and $\triangle YXX^\prime$. Clearly $\triangle WXX^\prime$ is inside the circle with diameter $WX$, and $\triangle YXX^\prime$ is inside the circle with diameter $XY$ (special case of inscribed angle theorem). Then $\triangle WXY$ is inside the union of those two circles.

Repeat for $\triangle YZW$.