A Rhombus Angle Trisection Result

Can you logically explain (prove) the following interesting little result from Posamentier, A. & Salkind, C. (1996, p. 4). Challenging Problems in Geometry. New York: Dover Publications?

"The trisectors of the angles of a rhombus ABCD are drawn. For each pair of adjacent angles, those trisectors that are closest to the enclosed side are extended until a point of intersection is established. The line segments connecting those points of intersection form a rectangle EFGH."

Drag points A, B or C to dynamically change the figure. (Note that the angle measures are in radians).

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A Rhombus Angle Trisection Result

1) Using suitable dynamic geometry, investigate what quadrilateral is formed if the angles of a parallelogram or kite are trisected in the same way? Can you logically explain (prove) your observations?

2) Using suitable dynamic geometry, investigate what quadrilateral is formed in the same way, if the angles of a rhombus, or the other quadrilaterals in 1), are divided into four, five, six or n equal parts? Can you logically explain (prove) your observations?

3) An interesting special case for a rhombus is when its angles are divided into four equal parts. Go to Rhombus Angle Quadrisection to investigate.

4) Can you generalize further to similar results as in 1) and 2) for certain hexagons, octagons, etc.?


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Michael de Villiers, modified 25 April 2011.