A line cut a parallelogram and a circle in half

Question

Given a circle and a parallelogram on the plane, show that there exists a line that cuts the area and the perimeter of the circle in half and cut the area of the parallelogram in half.

I know that the line exists, since one can just connect the center of the circle and the center of the parallelogram (intersection of diagonals). But my question is, when does there exists more than one line satisfying the condition? Maybe when the parallelogram is a rectangle(or a square)?

Note that I haven't found any configuration with more than one line satisfy the given condition.

Robert Z · Accepted Answer · 2019-02-03T09:45:46.023

1

There is just one line if the parallelogram and the circle have distinct centers.

Hint. A line cuts the area of the parallelogram in half if and only if the line goes through the center. Assume that there is a line with such property which does not go through the center. Then there is a distinct parallel line that goes through the center which cuts the area of the parallelogram in half. Find a contradiction.

P.S. This is a related problem (about existence and not uniqueness): The Hole in One Pizza

edited Feb 03 '19 at 09:45

answered Feb 03 '19 at 09:39

Robert Z

145,942

What if the perimeter of the parallelogram must be the same instead of the area? – abc... Feb 03 '19 at 11:08
@abc... The same argument works also for the perimeter of the parallelogram. – Robert Z Feb 03 '19 at 11:14
Robert Z thank you! :) – abc... Feb 03 '19 at 11:17

A line cut a parallelogram and a circle in half

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