Triangle of midpoints/Iteration/Convergence/Exercise

For a triangle ${}\triangle =(A,B,C)$, its triangle of midpoints is given by the vertices ${}{\frac {1}{2}}(A+B),{\frac {1}{2}}(A+C),{\frac {1}{2}}(B+C)$. This construction yields a recursively defined sequence of triangles ${}\triangle _{n}$, where ${}\triangle _{1}=\triangle$, and ${}\triangle _{n+1}$ is the triangle of midpoints of ${}\triangle _{n}$. Let ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ be a sequence in ${}\mathbb {R} ^{2}$ with ${}x_{n}\in \triangle _{n}$ for all ${}n\in \mathbb {N} _{+}$. Show that this sequence converges, and determine its limit.

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