# A Quadrilateral With Equal Opposite Sides And Angles:

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A Mathematical Droodle

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Copyright © 1996-2018 Alexander Bogomolny

The applet provides an illustration to a construction suggested by Nathan Bowler of a simple quadrilateral with a pair of equal opposite angles and a pair of equal opposite sides but which is not a parallelogram. (The construction came in response to a question posted at one of the old CTK Exchange forums.)

What if applet does not run? |

Let ABC be isosceles with AB = AC. Pick D on BC. Let C' be the reflection of C in the perpendicular bisector of AD. ABDC' has two opposite sides the same length and two opposite angles equal but is not a parallelogram if D isn't the midpoint of AB. This construction gives all such quadrilaterals.

Since as D glides over BC, neither AB nor ABC change, there is a continuum of quadrilaterals with the same pair of equal opposite angles and the same pair of equal opposite sides.

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Copyright © 1996-2018 Alexander Bogomolny