MathDB

Consider a sequence

P_{1}, P_{2}, \ldots

of points in the plane such that

P_{1}, P_{2}, P_{3}

are non-collinear and for every

n \geq 4, P_{n}

is the midpoint of the line segment joining

P_{n-2}

and

P_{n-3}

. Let

L

denote the line segment joining

P_{1}

and

P_{5}

. Prove the following:
[*] The area of the triangle formed by the points

P_{n}, P_{n-1}, P_{n-2}

converges to zero as

n

goes to infinity.
[*] The point

P_{9}

lies on

L

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