Math CentralQuandaries & Queries


Question from Apollo:

Prove analytically that if ABCD is a parallelogram in which points P and Q trisects the diagonal AC, then BPDQ is a parallelogram.

Hi Apollo,

I drew a diagram in the Cartesian plane of what you described. I put $A$ at the origin, $B$ on the X-axis with coordinates (0,b), and $C$ at (s, t).


Since $AB$ is parallel to $CD,$ and $BC$ is parallel to $AD,$ you can determine the coordinates of $C$ and $D.$

What are the coordinates of $P$ and $Q?$

To show that $BPDQ$ is a parallelogram find the slopes of its four sides.



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