Proving Bisected Opposite Sides of Quadrilaterals

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The discussion focuses on proving that the line segments joining the midpoints of opposite sides of a quadrilateral bisect each other. While the proof is straightforward for parallelograms and rectangles, challenges arise when attempting to apply the same logic to other quadrilaterals like trapezoids. The user seeks guidance on using coordinate geometry to establish the necessary expressions and equations for midpoints and line segments. By analyzing an arbitrary quadrilateral defined by specific points, the user aims to demonstrate that the midpoints of the segments intersect. The conversation concludes with the user expressing gratitude after gaining clarity on the topic.
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The line segments joining the midpoints of the opp. sides of a quadrilateral biset each other.


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The Attempt at a Solution


When I try to prove this for a parallelogram or a rectangle, it seems really obvious. Any number of ways can show this, by using triangles that are similar (using values of x1 and x2.. y1, y2), or simply just for the fact that since the opp are parallel that they have to be equal because if they weren't, the other set of lines could not be parallel. However, I can't find a way to PROVE that it would work for some other quadrilaterals like trapezoids and some of the funky looking ones. I would appreciate a little bump in the right direction, especially since we don't really deal with proofs in my class, its mostly applied math.
 
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Use co-ordinate geometry methods:

An arbitrary quadrilateral is described the the points (0,0) (0,a) (b,c) (d,e). Find expressions for the midpoints, and hence the equations of the line segments. Also find the midpoint of line segments and use the equations to show the other line passes through as well.
 
Thank you gib. I got it now.
 

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