Let
![A_r,B_r, C_r](/media/m/8/d/b/8db23613484f9c60b636311e056c1935.png)
be points on the circumference of a given circle
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
. From the triangle
![A_rB_rC_r](/media/m/6/b/3/6b3b1d79292f7ee1c4378eaf2ea064ed.png)
, called
![\Delta_r](/media/m/d/f/5/df5eaeab1cd3b1c0143c989bfd1237fa.png)
, the triangle
![\Delta_{r+1}](/media/m/b/7/e/b7e946337de1a9a21f9318257bca405b.png)
is obtained by constructing the points
![A_{r+1},B_{r+1}, C_{r+1}](/media/m/9/e/a/9ea0daa536a98081f1792a31558a7074.png)
on
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
such that
![A_{r+1}A_r](/media/m/c/d/6/cd689d078aeeb85b50399edd00f182fb.png)
is parallel to
![B_rC_r](/media/m/2/b/1/2b101ff1c41ead6dc7037fbdf1e110ac.png)
,
![B_{r+1}B_r](/media/m/4/e/6/4e6f1fc5ce0532d88eddc53237a493f5.png)
is parallel to
![C_rA_r](/media/m/b/0/9/b090e4787d410802d635d1e2a0de6814.png)
, and
![C_{r+1}C_r](/media/m/9/1/4/914ebd32d68844d97ffda968301fa9a6.png)
is parallel to
![A_rB_r](/media/m/8/6/d/86d0ffd857263318c2d6667cd3d721d2.png)
. Each angle of
![\Delta_1](/media/m/c/4/8/c48a1920a0f372e188f703b229112d5e.png)
is an integer number of degrees and those integers are not multiples of
![45](/media/m/4/6/f/46f8038988dda9b6ded9facde2a5e512.png)
. Prove that at least two of the triangles
![\Delta_1,\Delta_2, \ldots ,\Delta_{15}](/media/m/0/f/7/0f7b1319b65abcc9d861ed3545159c15.png)
are congruent.
%V0
Let $A_r,B_r, C_r$ be points on the circumference of a given circle $S$. From the triangle $A_rB_rC_r$, called $\Delta_r$, the triangle $\Delta_{r+1}$ is obtained by constructing the points $A_{r+1},B_{r+1}, C_{r+1}$on $S$ such that $A_{r+1}A_r$ is parallel to $B_rC_r$, $B_{r+1}B_r$ is parallel to $C_rA_r$, and $C_{r+1}C_r$ is parallel to $A_rB_r$. Each angle of $\Delta_1$ is an integer number of degrees and those integers are not multiples of $45$. Prove that at least two of the triangles $\Delta_1,\Delta_2, \ldots ,\Delta_{15}$ are congruent.