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Poncelet's Coaxal Theorem


PonceletsCoaxalTheorem

If a cyclic quadrilateral ABCD is inscribed in a circle c_1 of a coaxal system such that one pair AC of connectors touches another circle c_2 of the system at P, then each pair of opposite connectors will touch a circle of the system (BD at P^' on c_2, AB at Q on c_3, CD at Q^' on c_3, DA at R on c_4, and CB at R^' on c_4), and the six points of contact P, P^', Q, Q^', R, and R^' will be collinear.

The general theorem states that if A_1, A_2, ..., A_n are any number of points taken in order on a circle of a given coaxal system so that A_1A_2, A_2A_3, ..., A_(n-1)A_n touch respectively (n-1) fixed circles X_1, X_2, ..., X_(n-1) of the system, then A_nA_1 must touch a fixed circle X_n of the system. Further, if A_1A_2, A_2A_3, ..., A_(n-1)A_n touch respectively any n-1 of the circles X_1, X_2, ..., X_n, then A_nA_1 must touch the remaining circle.


See also

Coaxal System

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References

Lachlan, R. "Poncelet's Theorem." §334-342 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 209-217, 1893.

Referenced on Wolfram|Alpha

Poncelet's Coaxal Theorem

Cite this as:

Weisstein, Eric W. "Poncelet's Coaxal Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PonceletsCoaxalTheorem.html

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