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. ProofLet the tangent at C intersect tangents AS and BS in points U and V, respectively. Theorem of similar triangles, applied twice, gives
which tells us that the quadrilateral SUFS is cyclic. x-Axis is simson of the focusAs 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 tangentsLambert'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
Parabola
|Activities| |Contact| |Front page| |Contents| |Geometry| |Store| Copyright © 1996-2012 Alexander Bogomolny |
| 40600995 |

