Two Squares and a Quadrilateral

A common vertex is shared by two squares A. Quadrilateral HIJK is formed using the segments joining the vertices and the midpoints of the diagonals of the square. What is the shape of the quadrilateral?

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Questions

1. Move the vertices of the two squares. Are your observations still the same? 
2. Make a conjecture about your observation. 
3. Prove your conjecture. 

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Reference
 http://www.ies.co.jp/math/java/geo/twosqsq/twosqsq.html

van Aubel's Theorem

Squares are placed outwardly at each side of quadrilateral ABCD. The center of the squares on the opposite sides are connected with segments (PN and MO). Move points A, B, C and D and observe what happens. Make a conjecture about your observations.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Van Aubel's Theorem

Given an arbitrary planar quadrilateral, place a square outwardly on each side, and connect the centers of opposite squares. Then van Aubel's theorem states that the two lines are of equal length and cross at a right angle.

Click here to download GGB file.