Let
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
be a convex plane quadrilateral and let
![A_1](/media/m/5/a/6/5a6ce1347567551c02239ff8d4ebee67.png)
denote the circumcenter of
![\triangle BCD](/media/m/3/b/1/3b147a09016f2df03b75355ec5030ba8.png)
. Define
![B_1, C_1,D_1](/media/m/6/4/8/6481b404346b869a1e772295276fa10a.png)
in a corresponding way.
(a) Prove that either all of
![A_1,B_1, C_1,D_1](/media/m/5/5/b/55bf1def0f50c199671f4e7348c5c4cc.png)
coincide in one point, or they are all distinct. Assuming the latter case, show that
![A_1](/media/m/5/a/6/5a6ce1347567551c02239ff8d4ebee67.png)
, C1 are on opposite sides of the line
![B_1D_1](/media/m/7/b/d/7bdc96821214a01a74a5c673ad1c9f60.png)
, and similarly,
![B_1,D_1](/media/m/4/2/9/42978b9f1373ba6be613137e689a7a56.png)
are on opposite sides of the line
![A_1C_1](/media/m/9/3/2/932752a76a906e245c34e4943fa92183.png)
. (This establishes the convexity of the quadrilateral
![A_1B_1C_1D_1](/media/m/8/e/a/8ea8991888072519f65b0a7e2f45de2d.png)
.)
(b) Denote by
![A_2](/media/m/a/2/5/a25c6dade4a684fc874981a7d65625f5.png)
the circumcenter of
![B_1C_1D_1](/media/m/b/b/b/bbb680be8195ad070334c9c80677bb1c.png)
, and define
![B_2, C_2,D_2](/media/m/0/1/2/01274b6137b61cc8d147630de76910b8.png)
in an analogous way. Show that the quadrilateral
![A_2B_2C_2D_2](/media/m/6/2/9/629cf266bd45fcf773bea434616f461c.png)
is similar to the quadrilateral
%V0
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 $ABCD.$