# Re: geometry proof

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Posted by Soroban on October 29, 2002 at 11:15:33:

In Reply to: geometry proof posted by jacob on October 29, 2002 at 07:44:43:

: How do i write a two column proof for this problem.
: Given: Isoceles trapezoid PQRS with bases PQ and SR
: and diagonals SQ and PR; W, X, Y ,Z are midpoints of sides PS, PQ, QR, and RS respectively.
: Prove: WXYZ is a rhombus.

Hello, Jacob!

Did you know this?
If we connect the midpoints of the sides of ANY quadrilateral
(even a "bent" one in 3-space), we get a parallelogram?

In triangle PRS, X and W are midpoints of PQ and PS.
Hence, XW || QS and XW = QS/2.

Similarly, in triangle RQS, YZ || QS and YZ = QS/2.

Therefore, WXYZ is a parallelogram (a pair of opposites sides
are parallel and equal).

Now, we must show that, say, XW = XY.
There are many ways to accomplish this.
For example, show that triangles PRS and QRS
are congruent, then PR = QS.

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