Circumcircle of Three Parabola Tangents

The circumcircle of a triangle formed by three tangents to a parabola, passes through the focus of the parabola.

This is known as Lambert's theorem.

Proof

Let the tangent at $C$ intersect tangents $AS$ and $BS$ in points $U$ and $V,$ respectively. Theorem of similar triangles, applied twice, gives

$\angle FSU = \angle FBS = \angle FVU,$

which tells us that the quadrilateral $SUFS$ is cyclic.

x-Axis is simson of the focus

As we saw earlier, $x$-axis is the pedal curve of the parabola with respect to its focus. In other words, $x$-axis consists of the feet of the perpendiculars from the focus to the tangents to the parabola. This means that $x$-axis is the simson of the focus with respect to the circumcircle of any three tangents to the parabola [Honsberger, p. 48].

Parabola from four tangents

Lambert's theorem suggests a construction of parabola from four tangents. Any three tangents determine a circle that passes through $F.$ Two such circles determine F uniquely. Reflections of $F$ in any two tangents produce two points on the directrix.

References

  1. H. Dörrie, 100 Great Problems Of Elementary Mathematics, Dover Publications, NY, 1965
  2. R. Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, MAA, 1995.
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