Question 5
I 5 (GBR 3)
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 $\triangle_r$, the triangle $\triangle_{r+1}$ 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 $\triangle_1$ is an integer number of degrees and those integers are not multiples of $45$. Prove that at least two of the triangles $\triangle_1,\triangle_2,\ldots,\triangle_{15}$ are congruent.
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 $\triangle_r$, the triangle $\triangle_{r+1}$ 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 $\triangle_1$ is an integer number of degrees and those integers are not multiples of $45$. Prove that at least two of the triangles $\triangle_1,\triangle_2,\ldots,\triangle_{15}$ are congruent.