Expert: Scotto - 12/17/2008

Question
Give a vector proof that the midpoints of the sides of a square are the vertices of a square.

Thanks in advance! :)

Answer
A point proof comes to mind first, so I'll start with that.

x-y coordinates
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A graph can be positioned in such a way that one side is on the
x-axis with the opposite side parallel to the x-axis, the thrid side is on the y-axis, and the fourth side is parallel to the y-axis.

Since all sides have the same lenght, we'll call it L.  The corners would be at (0,0), (L,0), (0,L), and (L,L).  The midpoints of the sides would be at (L/2,0), (0, L/2), (L/2,L), and (L,L/2).
The connections of each of these corners would form a new square.  This is for two reasons.  They all have the same length, since they are each a hypoteneuse on a right triange with sides L/2.  Since they are hypoteneuse on a right triangle with equal sides, the next triangle around the square can be seen to have one of the legs flipped 180°, so the line is flipped 90°.  Therefore since they have the same length and are at 90° to each other, a new square is formed.


vector space
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It starts something like this.  Let the square again be positoned with a corner at the origin.  Two of the sides are perpendicular vectors, both with length L.  The other two sides are also vectors that originate at the endpoint of each of the first two at right angles.

I believe it is known that if the midpoint of each of the vectors is connected to the midpoint of the two adjacent vectors, it can be shown that they are of the same length since each new side can be rotated with out changing the length and it will fall into the place of one of the adjacent sides.

Since we know that two sides start at the same point and have the same length, it can be shown that they intersect the midpoint line at the same angle.  Using this, we can see that each of the four trianlges formed by two of the sides and a diagonal in the new figure are equal.

Since they are equal, it can also be shown that these angles each bisect equal angles of a square.  It is know that for any quadrilateral, the sum of the angles is (4-2)180° = 360°.  Since there are four equal angles that add to 360°, each must be a 90° angle.  Since the sides are all equal, it is a square.