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 A' | = | -sin ϕ:sin(C + ϕ):sin(B + ϕ) B' | = | sin(C + ϕ): - sin ϕ:sin(A + ϕ) C' | = | sin(B + ϕ):sin(A + ϕ): - sin ϕ, where ϕ is the base angle of the isosceles triangle. Kiepert showed that the lines connecting the vertices of the given triangle and the corresponding peaks of the isosceles triangles concur.
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