# Parabolic Reciprocity

### What Might This Be About?

### Problem

Given two parabolas $(V,UW)$ with the focus at $V$ and $UW$ as the directrix, and $(W,UV)$ with $W$ as the focus and $UV$ the directrix.

Prove that the common tangent to the two parabolas passes through $U.$

### Proof

Let $t$ be the tangent to $(V,UW)$ from $U$ and let $T$ be the point of tangency. Let $T'$ be the foot of the perpendicular from $T$ to $UW.$

By the tangent/bisector property, $\angle UTV=\angle UTT'.$ By the focal definition of parabola, $TV=TT'.$ It follows that $\Delta UTV=\Delta UTT',$ implying $\angle TUV=\angle TUT'.$ Thus $t$ is the angle bisector of $\angle VUW.$

The situation is absolutely symmetric with respect two the two parabolas, meaning that the tangent to $(W,UV)$ from $U$ also bisects $\angle VUW;$ thus, the two bisectors coincide.

### Acknowledgment

The above statement is a lemma by Emmanuel José García posted at his blog site.

### Conic Sections > Parabola

- The Parabola
- Archimedes Triangle and Squaring of Parabola
- Focal Definition of Parabola
- Focal Properties of Parabola
- Geometric Construction of Roots of Quadratic Equation
- Given Parabola, Find Axis
- Graph and Roots of Quadratic Polynomial
- Greg Markowsky's Problem for Parabola
- Parabola As Envelope of Straight Lines
- Generation of parabola via Apollonius' mesh
- Parabolic Mirror, Theory
- Parabolic Mirror, Illustration
- Three Parabola Tangents
- Three Points on a Parabola
- Two Tangents to Parabola
- Parabolic Sieve of Prime Numbers
- Parabolic Reciprocity
- Parabolas Related to the Orthic Triangle

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