A path from $A$ to $B$ consists of several piecewise straight segments (where $b \neq A$). A flea starts from point $P$ and for each segment, it performs the following operation:

If the segment starts at $X$ and ends at $Y$, and the flea is currently at $Z$, the flea will jump to the point $Z'$ such that $XZYZ'$ is a parallelogram (in that order).

The flea starts from $P$ and performs the operation on the segments sequentially until it is done with the last segment, where the flea is now in $Q$. Prove that $APBQ$ is a parallelogram.

## Thursday, November 12, 2009

### Path and Parallelogram

Labels:
Combinatorics,
flea,
Geometry,
jump,
parallelogram,
path,
piecewise,
segment

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