## 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. What kind of quadrilateral EFGH is formed by the line segments connecting the points of intersection as shown?"

Drag points A, B or C to dynamically change the figure. Click on the Button to display the measured lengths of the sides of the formed figure. What do you notice?

#### .sketch_canvas { border: medium solid lightgray; display: inline-block; } A Rhombus Angle Trisection Result

Investigate
1) Using suitable dynamic geometry, investigate what quadrilateral is formed if the angles of a rectangle, 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.? Especially have a look at these generalizations of rectangles and rhombi to higher polygons at Semi-regular Angle-gons and Side-gons.

Read my 2011 paper in Mathematics Teaching and Learning & Teaching Mathematics at Simply Symmetric.

Created by Michael de Villiers, modified 25 April 2011; 19 April 2021.