Steiner deltoid
- Author:
- Zoltán Kovács
- Topic:
- Parametric Curves
The Steiner deltoid is a curve which is named after Jakob Steiner who identified Euler's deltoid curve as the envelope of a family of the Simson lines of a triangle.
Precisely speaking, let us consider triangle ABC and its circumcircle. (Check Labels in the applet.) Let us choose an arbitrary point P on the circle and the three closest points E, F, G to P on lines AB, AC, and BC. Now E, F and G are collinear and the line they define is called the Simson line of triangle ABC.
The trace of the Simson line defines an envelope which is a geometric shape having the property that each member of the family of Simson lines are tangents to it.
GeoGebra's Envelope command can compute and visualize this envelope. In a web browser it is also possible to show the envelope, however it takes a while to manipulate on the appropriate algebraic equation system and get the proper curve.
You can try to drag the light blue point (which plays the role of P in the description above). Also you can change the position of the initial triangle by dragging the dark blue points keeping them on grid points.
Note that the initial triangle is always inside the deltoid curve.