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Let point M be outside of △ABC. Point N is the reflected image of M about the midpoint of segment AB . Point K is the reflected image of N about the midpoint of segment BC , and point K is the reflected image of L about the midpoint of segment AC . Prove that point A is the midpoint of segment ML .

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sqdancefan

Explanation:

Define points D, E, F as the midpoints of AB, BC, and AC, respectively. Point D is the midpoint of both AB and MN, so AMBN is a parallelogram, and side AM is parallel to and congruent with side NB.

Point E is the midpoint of both BC and NK, so BNCK is a parallelogram with side NB parallel and congruent to side CK, and by the transitive property of congruence, also to segment AM.

Point F is the midpoint of both AC and KL, so AKCL is a parallelogram with side CK parallel and congruent to side LA. By the transitive properties of congruence and of parallelism, sides AM, NB, CK, and LA are all congruent and parallel. Since AM and LA are congruent to one another and parallel, and share point A, point A must be their midpoint.

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