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.
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.
- 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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