TY - JOUR

T1 - The six circles theorem revisited

AU - Ivanov, Dennis

AU - Tabachnikov, Serge

N1 - Funding Information:
Most of the experiments that inspired this note and of the drawings were made in GeoGebra. The second author was supported by the NSF grant DMS-1105442. We are grateful to the referees for their criticism and advice.
Publisher Copyright:
© The Mathematical Association of America.

PY - 2016/9/1

Y1 - 2016/9/1

N2 - The six circles theorem of C. Evelyn, G. Money-Coutts, and J. Tyrrell concerns chains of circles inscribed into a triangle: the first circle is inscribed in the first angle, the second circle is inscribed in the second angle and tangent to the first circle, the third circle is inscribed in the third angle and tangent to the second circle, and so on, cyclically. The theorem asserts that if all the circles touch the sides of the triangle, and not their extensions, then the chain is 6-periodic. We show that, in general, the chain is eventually 6-periodic but may have an arbitrarily long pre-period.

AB - The six circles theorem of C. Evelyn, G. Money-Coutts, and J. Tyrrell concerns chains of circles inscribed into a triangle: the first circle is inscribed in the first angle, the second circle is inscribed in the second angle and tangent to the first circle, the third circle is inscribed in the third angle and tangent to the second circle, and so on, cyclically. The theorem asserts that if all the circles touch the sides of the triangle, and not their extensions, then the chain is 6-periodic. We show that, in general, the chain is eventually 6-periodic but may have an arbitrarily long pre-period.

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U2 - 10.4169/amer.math.monthly.123.7.689

DO - 10.4169/amer.math.monthly.123.7.689

M3 - Article

AN - SCOPUS:84981311925

SN - 0002-9890

VL - 123

SP - 689

EP - 698

JO - American Mathematical Monthly

JF - American Mathematical Monthly

IS - 7

ER -