[This is the third and final post in this series on triangle geometry. See the previous posts on Morley’s theorem and the 9-point circle.]

For our final exploration in this series, let’s again begin with our triangle *ABC* and a point *P* on the **circumcircle** of the triangle, i.e., the circle through the three vertices. If we drop *P* directly onto the three lines of the triangle at right angles^{[1]}, then by coincidence, these three points lie on a single line, called the **Simson line** of point *P*.

Just for fun, let’s draw point *Q* diametrically opposite from *P* on the circumcircle, and let’s also look at *Q*‘s Simson line. How do these two Simson lines interact? Somewhat surprisingly, these lines intersect at right angles. The phenomenal part is that this point of intersection lies on the 9-point circle!

Furthermore, as *P* and *Q* move around the circumcircle, the intersection of their Simson lines moves around the 9-point circle at twice the speed and in the opposite direction:

This story gets even more unbelievable when we look at how the Simson lines move in this animation. As it turns out, the Simson lines trace a curve in the shape of a **deltoid**, which is like an equilateral triangle with curved sides^{[2]}. The deltoid traced here is called **Steiner’s deltoid**.

And finally, here’s an incredible fact that ties everything together: If we draw the equilateral triangle around this deltoid, then the edges are parallel to the edges of the equilateral Morley triangle(s).

Holy Morley!

### Notes

- Note that we may have to extend the lines beyond the triangle. [↩]
- Specifically, a deltoid is the shape that results when you roll a circle inside a circle three-times larger and trace the path of a single point. See Wikipedia:Deltoid_curve for more information. [↩]