HyperbolaRelatedToEquilateral

Hyperbola related to equilateral

Consider a circle and a fixed direction. For example a circle (c) centered at the origin and the direction of the y-axis. Let A be an intersection point of the circle with the x-axis and consider a moving point B on the circle.
From B draw parallel to the y-axis intersecting the circle at a second point D. The intersection point E of lines AB and the tangent t_D to the circle at D describes a hyperbola.
This hyperbola has c as its auxiliary circle and passes through A and its diametral point I.
Further its asymptotes form an angle of pi/3.

The proof is an easy calculation in cartesian coordinates. In the case the circle has unit radius, the resulting equation of E is 3x² - y² = -3. For circles with other radii the corresponding hyperbola is homothetic to this one. In particular the asymptotes remain the same and are parallel to the sides of the equilateral inscribed in the circle and with one vertex at A.
Notice some other properties of this hyperbola.
1] DA bisects angle BDE.
2] Triangle AED is similar to DEB.
3] The hyperbola is the locus of intersections of BA with the line symmetric to BD w.r. to AD.
Remark The same hyperbola can be viewed as envelope of lines in the following way.
Consider the equilateral triangle CNS and draw two parallels from {N,S} intersecting the opposite sides respectively at points {N',S'}. Line N'S' is always tangent to the hyperbola.
The proof for this is given in HyperbolaPropertyParallels.html .

Hyperbola related to equilateral

See Also