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

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 3x

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.

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 .

HyperbolaAsymptotics.html

HyperbolaPropertyParallels.html

Produced with EucliDraw© |