# Thread: Proof: line segments joining midpoints of opposite sides of quadrilateral bisect

1. ## Proof: line segments joining midpoints of opposite sides of quadrilateral bisect

Hello, I am reviewing math over the summer and am having trouble with this problem:

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

The trouble is that this isn't from a geometry textbook, it's from the precal review section of a calculus textbook. The section it appears in covers only the Pythagorean theorem, the distance formula, and the midpoint formula. Hence I assume I'm supposed to prove this algebraically, with not much more information than that. How would I do this?

2. ## Re: Proof: line segments joining midpoints of opposite sides of quadrilateral bisect

Hello, Ragnarok!

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

Code:
           A      P      B
o  *  o  *  o
*              *
S *                 *  Q
o                    o
*                       *
*                          *
o  *  *  *  *  o  *  *  *  *  o
D               R               C
We have quadrilateral $\displaystyle ABCD$ with midpoints $\displaystyle P,Q,R,S.$
Draw $\displaystyle PQ,\,QR,\,RS,\,SP.$
Draw diagonals $\displaystyle AC$ and $\displaystyle BD.$

Theorem: The line segment joining the midpoints of two sides of a triangle
. . . . . . . .is parallel to and one-half the length of the third side.

$\displaystyle \text{In }\Delta ABC:\;PQ \parallel AC\,\text{ and }\,\overline{PQ} \,=\,\tfrac{1}{2}\overline{AC}$

$\displaystyle \text{In }\Delta ADC:\;RS \parallel AC\,\text{ and }\,\overline{RS} \,=\,\tfrac{1}{2}\overline{AC}$

. . $\displaystyle \text{Hence: }\,PQ \parallel RS\,\text{ and }\,\overline{PQ} \,=\,\overline{RS}$

Theorem: If two sides of a quadrilateral are parallel and equal,
. . . . . . . .the quadrilateral is a parallelogram.'

$\displaystyle \text{Hence, }PQRS\text{ is a parallelogram.}$

$\displaystyle \text{Fact: the diagonals of a parallelogram bisect each other }\;\;\hdots \;\;\text{Q.E.D.}$

3. ## Re: Proof: line segments joining midpoints of opposite sides of quadrilateral bisect

Originally Posted by Ragnarok
The trouble is that this isn't from a geometry textbook, it's from the precal review section of a calculus textbook. The section it appears in covers only the Pythagorean theorem, the distance formula, and the midpoint formula. Hence I assume I'm supposed to prove this algebraically, with not much more information than that.
Let the quadrilateral vertices have coordinates (x1, y1), ..., (x4, y4). Using the midpoint formula, find the midpoints of the sides and then the midpoints of the segments joining the midpoints of the opposite sides. Show that the latter two midpoints coincide.

4. ## Re: Proof: line segments joining midpoints of opposite sides of quadrilateral bisect

Thank you both so much! This question was driving me crazy! I appreciate the geometric proof but it seemed too subtle and involved prior geometry knowledge. emakarov, I started it your way but then got confused and gave up. Thanks for clarifying it!

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# prove that the line segments joining the opposite sides of a quadrilateral bisect each other

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