Prove that the line segments joining the midpoints of the opposite sides of a quadrilateral bisect each other.

Question

ANALYTIC GEOMETRY:

Prove that the line segments joining the midpoints of the opposite sides of a quadrilateral bisect each other.

Answer 1

If I plot the four vertices of a quadrilateral inside of a coordinate grid, I can take the midpoints of each of these to get a new set of points. If I take the midpoint of one pair that lies on opposite sides of the quadrilateral, I'll get the same coordinate as I would if I did this to the other pair. 

 

Draw a quad in a plane and go around in a circle labeling the points to be A, B, C, and D.

 

Assign each of these a coordinate like (x_A, y_A) and (x_B, y_B) and (x_C, y_C) and lastly (x_D, y_D).

 

The midpoint of AB will be ((x_A + x_B)/2, (y_A + y_B)/2). 

 

Find the midpoints of BC, CD, and DA doing the same thing.

 

Now notice that if you take the midpoint of the midpoints of AB and CD you will get the same thing that you get if you take the midpoint of the midpoints of BC and DA. 

 

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