Given a conic section $C$ it is easy to prove analytically (or algebraically) that there is a unique tangent to $C$ in each point. Is there a simple synthetic proof of this fact? References are also welcome.
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What is the synthetic definition of "tangent line at a point"? – Crostul Jan 11 '16 at 21:00
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What is a synthetic proof – ClassicStyle Jan 11 '16 at 21:02
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By synthetic you mean that which includes all conics? If so slope of tangent is found by direct diierentiation. – Narasimham Jan 11 '16 at 21:05
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A synthetic proof is a classical geometric proof a la Euclid. – Jan 11 '16 at 21:26
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And the classical definition of a tangent to a conic is a line that intersects the conic in a single point such that the conic is on one side of the line (with a slight modification in the case of hyperbolas). – Jan 11 '16 at 21:28
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Consider 2 planes: one tangent to a cone in the point of interest, another is the secant plane. The intersection is a line. Prove it is tangent to conic. Very straightforward. – user58697 Jan 11 '16 at 21:41
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You have constructed a unique tangent. I do not see how this is supposed to show that there cannot be two of them. – Jan 12 '16 at 07:25
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Prove that a tangent must belong to both planes. That makes it unique. – user58697 Jan 12 '16 at 13:53