Let

be a convex plane quadrilateral and let

denote the circumcenter of

. Define

in a corresponding way.
(a) Prove that either all of

coincide in one point, or they are all distinct. Assuming the latter case, show that

, C1 are on opposite sides of the line

, and similarly,

are on opposite sides of the line

. (This establishes the convexity of the quadrilateral

.)
(b) Denote by

the circumcenter of

, and define

in an analogous way. Show that the quadrilateral

is similar to the quadrilateral
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Let $ABCD$ be a convex plane quadrilateral and let $A_1$ denote the circumcenter of $\triangle BCD$. Define $B_1, C_1,D_1$ in a corresponding way.
(a) Prove that either all of $A_1,B_1, C_1,D_1$ coincide in one point, or they are all distinct. Assuming the latter case, show that $A_1$, C1 are on opposite sides of the line $B_1D_1$, and similarly,$B_1,D_1$ are on opposite sides of the line $A_1C_1$. (This establishes the convexity of the quadrilateral $A_1B_1C_1D_1$.)
(b) Denote by $A_2$ the circumcenter of $B_1C_1D_1$, and define $B_2, C_2,D_2$ in an analogous way. Show that the quadrilateral $A_2B_2C_2D_2$ is similar to the quadrilateral $ABCD.$