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January 1, 2019

What if a Simson Line moves!

This is the story of a deltoid.

It is known as Steiner Hypocycloid.

It looks like a curved sided triangle. And it is traced out by something called a Simson line.

Let us start the story from the beginning.

We have a triangle ABC. We draw its circumcircle. And we pick any point P on the circumcircle.

Next drop, perpendiculars from P to the sides of the triangle. We may have to extend the sides to mark the feet of the perpendiculars.

Alright, so you have this picture now. Suppose the feet of the perpendiculars are A1, B1, and C1.

It is a well-known theorem that these feet of perpendiculars, that is A1, B1, and C1 are collinear. That is they lie on the same straight line. In fact, this is not hard to prove. Why don’t you prove this using basic angle chasing?

This line on which A1, B1, and C1 are located, is known as the Simson Line. Apparently, someone else called Wallace proved this first.

Next, we will do an experiment with this Simson Line. We will move the point P along the circumference of this circle.

As P runs along the circumference the Simson line also moves and it envelops or creates a very beautiful symmetric structure which is looking like a curved triangle. This is precisely the Steiner Hypocycloid.

This was first discovered by Steiner in the middle of 19th century.

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