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Conies are often presented through the following definition: The conic Γ with focus F, directrix l and eccentricity e is the set of points Y of the Euclidean plane E2 such that d(Y, F) = e d(Y, l). Of course, the main interest in this focus-directrix property is to allow a unified approach to the three conies. Surprisingly, a simple construction of Γ given F, l, e seldom follows. The aim of this note is to fill the gap by describing … just one construction for the three conies! Readers are strongly encouraged to try out this construction, either by hand, using a ruler and a set-square, or by using a computer package. As is to be expected, the tangents to Γ are also easily obtained. Moreover, the transformation underlying the whole process leads to some properties of the conies in a simple and elegant way. A few examples will be given.
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