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Double Contact Theorem


DoubleContactTheorem

If S_1, S_2, and S_3 are three conics having the property that there is a point X, not on any of the conics, lying on a common chord of each pair of the three conics (with the chords in question being distinct), then there exists a conic S_4 that has a double contact with each of S_1, S_2, and S_3 (Evelyn et al. 1974, p. 18).

The converse of the theorem states that if three conics S_1, S_2, and S_3 all have double contact with another S_4 then each two of S_1, S_2, and S_3 have a "distinguished" pair of opposite common chords, the three such pairs of common chords being the pairs of opposite sides of a complete quadrangle (Evelyn et al. 1974, p. 19).

The dual theorems are stated as follows. If three conics are such that, taken by pairs, they have couples of common tangents intersecting at three distinct points on a line (that is not itself a tangent to any of the conics), then (a) the conics have this property in four different ways, and (b) the conics all have double contact with a fourth. And, conversely, if three conics each have double contact with a fourth, then certain of their common tangents intersect by pairs at the vertices of a complete quadrilateral (Evelyn et al. 1974, p. 22).

A degenerate case of the theorem gives the result that the six similitude centers of three circles taken by pairs are the vertices of a complete quadrilateral (Evelyn et al. 1974, pp. 21-22).


See also

Conic Section, Similitude Center

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References

Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. "The Double-Contact Theorem." §2.3 in The Seven Circles Theorem and Other New Theorems. London: Stacey International, pp. 18-22, 1974.

Referenced on Wolfram|Alpha

Double Contact Theorem

Cite this as:

Weisstein, Eric W. "Double Contact Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DoubleContactTheorem.html

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