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A Quadrilateral With Equal Opposite Sides And Angles: What Is This About?
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Copyright © 1996-2008 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 CTK Exchange forums.)
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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.
Copyright © 1996-2008 Alexander Bogomolny
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