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In geometry, Monge's theorem, named after Gaspard Monge, states that for any three circles in a plane, none of which is inside one of the others, the intersection points of each of the three pairs of external tangent lines are collinear.

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  • Monge's theorem
  • Twierdzenie Mongego
  • Teorema de Monge
  • Теорема Монжа
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  • Oznaczmy promienie okręgów przez odpowiednio r1,r2,r3, a środki przez O1,O2,O3. Niech ponadto Zij będzie przecięciem stycznych zewnętrznych do okręgów Oi i Oj. Ponieważ promienie prostopadłe do stycznej są równoległe, więc z twierdzenia Talesa mamy równości: . Ponieważ , więc z twierdzenia Menelaosa dla trójkąta punkty Z12,Z23,Z31 są współliniowe.
  • Na geometria, o teorema de Monge afirma que para quaisquer três círculos de um plano, os três pontos de interseção, de três pares de retas tangentes externas, serão colineares. Como condição de existência, nenhum dos círculos poderá estar situado por completo no interior de outro.
  • Теорема Мо́нжа — теорема о трёх окружностях, сформулированная Жаном Д’Аламбером и доказанная Гаспаром Монжем. Теорема. Для трёх произвольных окружностей, каждая из которых не лежит целиком внутри другой, точки пересечения общих внешних касательных к каждой паре окружностей лежат на одной прямой.
  • In geometry, Monge's theorem, named after Gaspard Monge, states that for any three circles in a plane, none of which is inside one of the others, the intersection points of each of the three pairs of external tangent lines are collinear.
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  • Oznaczmy promienie okręgów przez odpowiednio r1,r2,r3, a środki przez O1,O2,O3. Niech ponadto Zij będzie przecięciem stycznych zewnętrznych do okręgów Oi i Oj. Ponieważ promienie prostopadłe do stycznej są równoległe, więc z twierdzenia Talesa mamy równości: . Ponieważ , więc z twierdzenia Menelaosa dla trójkąta punkty Z12,Z23,Z31 są współliniowe.
  • Na geometria, o teorema de Monge afirma que para quaisquer três círculos de um plano, os três pontos de interseção, de três pares de retas tangentes externas, serão colineares. Como condição de existência, nenhum dos círculos poderá estar situado por completo no interior de outro.
  • In geometry, Monge's theorem, named after Gaspard Monge, states that for any three circles in a plane, none of which is inside one of the others, the intersection points of each of the three pairs of external tangent lines are collinear. For any two circles in a plane, an external tangent is a line that is tangent to both circles but does not pass between them. There are two such external tangent lines for any two circles. Each such pair has a unique intersection point in the extended Euclidean plane. Monge's theorem states that the three such points given by each circle are always in a straight line. In the case of two of the circles being of equal size, the two external tangent lines are parallel. In this case Monge's theorem asserts that the other two intersection points must lie on a line parallel to those two external tangents. In other words, if the two external tangents are considered to intersect at the point at infinity, then the other two intersection points must be on a line passing through the same point at infinity, so the line between them takes the same angle as the external tangent.
  • Теорема Мо́нжа — теорема о трёх окружностях, сформулированная Жаном Д’Аламбером и доказанная Гаспаром Монжем. Теорема. Для трёх произвольных окружностей, каждая из которых не лежит целиком внутри другой, точки пересечения общих внешних касательных к каждой паре окружностей лежат на одной прямой.
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