# 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

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

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