Multiplying and Dividing Curves by Points
نویسندگان
چکیده
Pointwise products and quotients, defined in terms of barycentric and trilinear coordinates, are extended to products P ·Γ and quotients Γ/P, where P is a point and Γ is a curve. In trilinears, for example, if Γ0 denotes the circumcircle, then P ·Γ0 is a parabola if and only if P lies on the Steiner inscribed ellipse. Barycentric division by the triangle center X110 carries Γ0 onto the Kiepert hyperbola Γ′; if P is on Γ0, then the point P ′ = P/X110 is the point, other than the Tarry point, X98, in which the line PX98 meets Γ′, and if Ω1 and Ω2 denote the Brocard points, then |P ′Ω1|/|P ′Ω2| = |PΩ1|/|PΩ2|; that is, P ′ and P lie on the same Apollonian circle with respect to Ω1 and Ω2.
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