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I'm stuck trying to solve the following exercise.

Let C be a conic in $RP^2$ and P a point on C. Give a geometric construction for the tangnt line to C at P only using a straight edge. (That is, one can draw the line through any two given points, but cannot measure distances.) Blockquote

My idea was to use self-polar triangles such that one of the vertices is the point P. This implies that one of the sides of the triangle is the polar line to P. But since P lies on the conic the polar line must also be the tangent of C at P. But so far I had no luck in finding the correct triangles.

Polymorph
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1 Answers1

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Given a point $A$ on a conic, choose other four points $B,C,D,E$.

Let $N=AE\cap BC$, $M=DE\cap AB$. Then $p=MN$ is the Pascal's line, thus by Pascal's theorem the tangent $a$ at $A$ meets $p$ at $H=p\cap CD$. Consequently, $a=AH$ is the required tangent.

Fabio Lucchini
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