TOPICS

Three Conics Theorem

If three conics pass through two given points and , then the lines joining the other two intersections of each pair of conics are concurrent at a point (Evelyn 1974, p. 15). The converse states that if two conics and meet at four points , , , and , and if and are chords of and , respectively, which meet on , then the six points lie on a conic. The dual of the theorem states that if three conics share two common tangents, then their remaining pairs of common tangents intersect at three collinear points.

If the points and are taken as the points at infinity, then the theorem reduces to the theorem that radical lines of three circles are concurrent in a point known as the radical center (Evelyn 1974, p. 15).

If two of the points and are taken as the points at infinity, then the theorem becomes that if two circles and pass through two points and on a conic , then the lines determined by the pair of intersections of each circle with the conic are parallel (Evelyn 1974, p. 15).

Conic Section, Four Conics Theorem, Radical Center

Explore with Wolfram|Alpha

More things to try:

References

Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. "The Three-Conics Theorem." §2.2 in The Seven Circles Theorem and Other New Theorems. London: Stacey International, pp. 11-18, 1974.

Referenced on Wolfram|Alpha

Three Conics Theorem

Cite this as:

Weisstein, Eric W. "Three Conics Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ThreeConicsTheorem.html