The Deltoid

Steven Dutch, Natural && Applied Sciences,University of Wisconsin - Green Bay
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The deltoid is a three- cusped hypocycloid formed by a circle of radius r rolling on the inside of a circle of radius 3r. The deltoid has another interesting property; it is the envelope of all the Simson's Lines of a triangle

Simson's Lines are generated thus; Pick any point on the circumference of the circumcircle of the triangle. from this point, drop perpendiculars to each of the three sides. extend the sides, if(necessary. The feet of the three perpendiculars will lie on a straight line.

This line is a Simson's Line (note; no 'p'). Altitudes of the triangle are Simson's Lines for some point (can you see why?) Sides of the triangle are Simson's Lines for some point (can you see why?) All the Simson's Lines of a given triangle trace out the deltoid.

The deltoid always has regular three-fold symmetry regardless of how asymetrical the generating triangle is.

There's one other surprising threefold symmetry here. if you trisect the angles of the generating triangle, locate the intersections of adjacent trisectors, and connect those three points, you get a triangle called Morley's Triangle. Morley's triangle is always equilateral regardless of the shape of the generating triangle. Morley's Triangle is oriented so its cusps point opposite the cusps of the deltoid. However, Morley's Triangle is not centered in the deltoid.

Proving this must have been a bear. Proving that an envelope results in a given curve is fairly complex calculus. The mathematics of Simson's line is complex. Putting the two together must have been a real beast.

The proof was established by J. Steiner in 1858. Steiner is also remembered for Steiner Chains, another very subtle geometrical tidbit. Steiner must have had a real love for geometry as well as difficult proofs.

It is very hard to predict just where a given line intersects the deltoid. The algorithm in this program calculates Simson's Line between the feet of the perpendiculars and extends it some distance beyond. Therefore the Simson's Lines extend beyond the deltoid

The Nine-Point Circle of the triangle is tangent to the deltoid. The Nine-Point Circle is so called because it passes through the midpoints of all three sides (an easy way to construct it), the feet of all three altitudes, and the midpoints of the line segments from each vertex of the triangle to the orthocenter (where the three altitudes meet). Its center is halfway between the circumcenter of the triangle and the orthocenter (intersection of all three altitudes) and its radius is half that of the circumcircle.

Major Step Minor Step Radius of Primitive Circle
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Created 14 February 2012, Last Update 18 March 12

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