matematicas visuales visual math

We know that Jacob Steiner (1796-1863) proved that the envelope of the Wallace-Simson lines is a curve that has three cuspidal points that is called the Steiner deltoid.

Steiner deltoid
The Simson-Wallace lines of a triangle envelops a curve called the Steiner Deltoid.

And a deltoid is one specific curve in the family of hypocycloid curves. A hypocycloid is a plane curve generated by the trace of a fixed point on a small circle that rolls without slipping within a larger circle. A deltoid is a hypocycloid in which the inner turning circle has a radius that is one third the radius of the outer circle.

In this page we explore the fact that the Steiner Deltoid is a hypocycloid. It is to say that it is generated by a circle rolling inside another circle.

Both circles are related with the Feuerbach circle or the nine-point circle, (green circle in the image). The radius of the nine-point circle is half the radius of the circumscribed circle. The exterior circle has the same center as the nine-point circle and its radius is three times the radius of the nine-point circle.

The rolling circle has the same radius as the Feuerbach circle and both are tangent.

Steiner Deltoid is a hypocycloid: | matematicasVisuales

When the small circle rolls without slipping generates the Steiner deltoid.

Steiner Deltoid is a hypocycloid: When the small circle rolls without slipping it generates the Steiner Deltoid | matematicasVisuales

The Steiner deltoid is tangent to the nine-point circle.

Steiner Deltoid is a hypocycloid: The Steiner Deltoid is tangent to the nine-point circle | matematicasVisuales

The three tangents to the deltoid at the three cuspides pass through the center of the Feuerbach circle.

Steiner Deltoid is a hypocycloid: The three tangents to the deltoid at the three cuspides pass through the center of the Feuerbach circle | matematicasVisuales

REFERENCES

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer.
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: John Wiley and sons, 1969.

MORE LINKS

Wallace-Simson lines
Each point in the circle circunscribed to a triangle give us a line (Wallace-Simson line)
Wallace-Simson lines | Demonstration
Interactive demonstration of the Wallace-Simson line.
Central and inscribed angles in a circle
Central angle in a circle is twice the angle inscribed in the circle.
Central and inscribed angles in a circle | Mostration | Case I
Interactive 'Mostation' of the property of central and inscribed angles in a circle. Case I: When the arc is half a circle the inscribed angle is a right angle.
Central and inscribed angles in a circle | Mostration | Case II
Interactive 'Mostation' of the property of central and inscribed angles in a circle. Case II: When one chord that forms the inscribed angle is a diameter.
Central and inscribed angles in a circle | Mostration | General Case
Interactive 'Mostation' of the property of central and inscribed angles in a circle. The general case is proved.
Drawing fifteen degrees angles
Using a ruler and a compass we can draw fifteen degrees angles. These are basic examples of the central and inscribed in a circle angles property.
Morley Theorem
The three points of intersection of the adjacent trisectors of the angles of any triangle are the vertices of an equilateral triangle (Morley's triangle)
John Conway's proof of Morley's Theorem
Interactive animation about John Conway's beautiful proof of Morley's Theorem