# A Quadrilateral With Equal Opposite Sides And Angles: What Is This About? A Mathematical Droodle

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 What if applet does not run?

Explanation

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.)

### This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

 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.