Three Conics Theorem


If three conics pass through two given points Q and Q^', then the lines joining the other two intersections of each pair of conics P_(ij)P_(ij)^' are concurrent at a point X (Evelyn 1974, p. 15). The converse states that if two conics E_2 and E_3 meet at four points Q, Q^', P_1, and Q_1, and if P_2Q_2 and P_3Q_3 are chords of E_3 and E_2, respectively, which meet on P_1Q_1, 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 Q and Q^' 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 P_(ij) and P_(ij)^' are taken as the points at infinity, then the theorem becomes that if two circles C_1 and C_2 pass through two points Q and Q^' on a conic E, then the lines determined by the pair of intersections of each circle with the conic are parallel (Evelyn 1974, p. 15).

See also

Conic Section, Four Conics Theorem, Radical Center

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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.

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Three Conics Theorem

Cite this as:

Weisstein, Eric W. "Three Conics Theorem." From MathWorld--A Wolfram Web Resource.

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