TY - JOUR
T1 - The six circles theorem revisited
AU - Ivanov, Dennis
AU - Tabachnikov, Serge
N1 - 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 -