The nine-point hyperbola was first discovered by E.F. Allen and his work was published in a volume of The American Mathematical Monthly in December, 1941. Allen was able to take the work that English Mathematician Frank Morley completed on the nine-point circle using complex numbers in his book Inverse Geometry (1933) and apply it to hyperbolas using split complex numbers using the equation zz* = 1 for hyperbolas in the split-complex plane.
The nine-point hyperbola was recalled by Isaak Yaglom when describing Minkowskian geometry in the conclusion of his book A Simple Non-Euclidean Geometry and its Physical Basis (1979). For Yaglom, a hyperbola is a Minkowskian circle. He says on page 193
- ...the midpoints of the sides of a triangle ABC and the feet of its altitudes (as well as the midpoints of the segments joining the orthocenter of ΔABC to its vertices) lie on a [Minkowskian] circle S whose radius is half the radius of the circumcircle of the triangle. It is natural to refer to S as the six- (nine-) point circle of the (Minkowskian) triangle ABC; if the triangle ABC has an incircle s, then the six- (nine-) point circle S of ΔABC touches its incircle s (Fig.173).
Starting out with a right hyperbola, we can find a focus and its reflection on the line of reflection y=x (Reflection Symmetry). Given these two points, we use a drafting compass to find two alternate points on the right hyperbola. When you draw a line through one of those points and the constructed foci point you get another point on the hyperbola.
A triangle can be constructed where all three vertices are the points previously constructed. Also the side constructed by the two points found using the drafting compass goes through the origin. Using this triangle, angle bisectors are constructed to produce the orthocenter. Also, by using this triangle the midpoint of each side is determined.
Once you have constructed the angle bisectors and the midpoints of each side, you now have all the points needed in order to construct the nine-point circle.
Similarly given a rectangular hyperbola, this same nine-point circle can be constructed using the same methods used above when complete; the rectangular nine-point hyperbola would look like this.
- Allen, E.F. On a Triangle Inscribed in a Rectangular Hyperbola, The American Mathematical Monthly, 1941. Vol. 48, No.10 pp. 675–681
- Bjørn Felsager(2004) Through the Looking Glass - A glimpse of Euclid’s twin geometry, the Minkowski geometry, ICME-10 Copenhagen.
- Hahn, Liang-shin. Complex Numbers and Geometry, The Mathematical Association of America, 1994. ISBN 0-88385-510-0