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In a rectangular hyperbola there are 3 points $A,B,C$ such that $ABC$ form a right triangle with right angle at $A$. Prove that the tangent at $A$ is perpendicular to $BC$. I am looking for a synthetic proof (which does not use co-ordinate geom.). (as one such proof is included at Tangents of Rectangular hyperbola )

I tried to do something with the reflection property of hyperbola (since the tangent will then become angle bisector), but i do not know how to connect that information to that of $BC$.

tarzanjunior
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1 Answers1

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This can be seen as a consequence of the following result:

A conic passing through the vertices of a triangle $ABC$ is an equilateral hyperbola if and only if it passes through the orthocenter of the triangle.

A proof of this can be found in the book Geometry of Conics, by Akopyan and Zaslavsky (Problem 19 on page 67).

In fact, consider a point $C'$ on the hyperbola, near to $C$. By the above theorem, the hyperbola passes through the orthocenter $O$ of $ABC'$, and line $AO$ is thus perpendicular to $BC'$. As $C'$ tends to $C$ angle $C'AB$ tends to a right angle, hence $O$ tends to $A$ and line $OA$ tends to the tangent at $A$. In the limit $C'\to C$ we then find that the tangent at $A$ is perpendicular to $BC$.

Intelligenti pauca
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