Kiepert Hyperbola

The Kiepert hyperbola is a hyperbola and triangle conic that is related to the solution of Lemoine's problem and its generalization to isosceles triangles constructed on the sides of a given triangle.

The vertices of the constructed triangles are given in trilinear coordinates by

			(1)
			(2)
			(3)

where is the base angle of the isosceles triangle.

Kiepert (1869) showed that the lines connecting the vertices of the given triangle and the corresponding peaks of the isosceles triangles concur. The trilinear coordinates of the point of concurrence are

(4)

The locus of this point as the base angle varies is given by the curve with trilinear coordinates

(5)

or equivalently

(6)

(Kimberling 1998, p. 237). This curve is a rectangular hyperbola known as the Kiepert hyperbola.

Kiepert center is Kimberling center , which has equivalent triangle center functions

			(7)
			(8)

(Kimberling 1998, p. 86).

The Kiepert hyperbola passes through Kimberling centers for , (triangle centroid), 4 (orthocenter), 10 (Spieker center; i.e., incenter of the medial triangle of ; Eddy and Fritsch 1994), 13 (first Fermat point), 14 (second Fermat point), 17 (first Napoleon point), 18 (second Napoleon point), 76 (third Brocard point), 83 (isogonal conjugate of the Brocard midpoint; Eddy and Fritsch 1994), 94, 96, 98 (Tarry point), 226, 262, 275, 321, 485 (outer vecten point), 486 (inner vecten point), 598, 671, 801, 1029, 1131, 1132, 1139 (inner pentagon point), 1140 (outer pentagon point), 1327, 1328, 1446, 1676, 1677, 1751, 1916, 2009, 2010, 2051, 2052, 2394, 2592, 2593, 2671 (first golden arbelos point), 2672 (second golden arbelos point), 2986, and 2996

A subset of these points is summarized in the following table together with their corresponding angles (Eddy and Fritsch 1994, p. 193; Kimberling 1998, pp. 176-178 and 237). Here, is the Brocard angle.

Kimberling center	triangle center
	vertex
	vertex
	vertex
	triangle centroid	0
	orthocenter
	Spieker center
	first Fermat point
	second Fermat point
	third Brocard point
	isogonal conjugate of the Brocard midpoint

Eddy and Fritsch (1994) also showed that the Kiepert hyperbola passes through the Spieker center.

The asymptotes of the Kiepert hyperbola are the Simson lines of the intersections of the Brocard axis with the circumcircle.

The isogonal conjugate of the Kiepert hyperbola is the Brocard axis, and the isotomic conjugate is the line through the triangle centroid and the symmedian point .

Writing the trilinear coordinates as

(9)

where is the distance to the side opposite of length and using the point-line distance formula with written as ,

(10)

where and gives the formula

			(11)
			(12)

Bringing this equation over a common denominator then gives a quadratic in and , which is a conic section (in fact, a hyperbola). The curve can also be written as , as varies over .