Введение
Геометрическое соотношение между отрезками, образованными прямой, пересекающей треугольник. В евклидовой геометрии теорема Менелая, названная в честь Менелая Александрийского, представляет собой утверждение о треугольниках в планиметрии. Пусть дан треугольник △ABC и секущая, пересекающая стороны BC, AC и AB в точках D, E и F соответственно, причем точки D, E и F отличны от A, B и C. Слабая версия теоремы утверждает, что
In Euclidean geometry, Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle △ABC, and a transversal line that crosses BC, AC, AB at points D, E, F respectively, with D, E, F distinct from A, B, C. A weak version of the theorem states that
where "| |" denotes absolute value (i. e., all segment lengths are positive). The theorem can be strengthened to a statement about signed lengths of segments, which provides some additional information about the relative order of collinear points. Here, the length is taken to be positive or negative according to whether A is to the left or right of B in some fixed orientation of the line; for example, is defined as having positive value when F is between A and B and negative otherwise. The signed version of Menelaus's theorem states
Equivalently,
Some authors organize the factors differently and obtain the seemingly different relation
but as each of these factors is the negative of the corresponding factor above, the relation is seen to be the same. The converse is also true: If points D, E, F are chosen on BC, AC, AB respectively so that
then D, E, F are collinear. The converse is often included as part of the theorem. (Note that the converse of the weaker, unsigned statement is not necessarily true.) The theorem is very similar to Ceva's theorem in that their equations differ only in sign. By re writing each in terms of cross ratios, the two theorems may be seen as projective duals.
где "||" обозначает абсолютную величину (то есть все длины отрезков положительны). Теорему можно усилить утверждением о знаковых длинах отрезков, что предоставляет дополнительную информацию об относительном порядке коллинеарных точек. Здесь длина считается положительной или отрицательной в зависимости от того, находится ли точка A слева или справа от точки B в некоторой фиксированной ориентации прямой; например, определяется как положительное значение, когда точка F лежит между точками A и B, и отрицательное – в противном случае. Знаковая версия теоремы Менелая утверждает:
In Euclidean geometry, Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle △ABC, and a transversal line that crosses BC, AC, AB at points D, E, F respectively, with D, E, F distinct from A, B, C. A weak version of the theorem states that
where "| |" denotes absolute value (i. e., all segment lengths are positive). The theorem can be strengthened to a statement about signed lengths of segments, which provides some additional information about the relative order of collinear points. Here, the length is taken to be positive or negative according to whether A is to the left or right of B in some fixed orientation of the line; for example, is defined as having positive value when F is between A and B and negative otherwise. The signed version of Menelaus's theorem states
Equivalently,
Some authors organize the factors differently and obtain the seemingly different relation
but as each of these factors is the negative of the corresponding factor above, the relation is seen to be the same. The converse is also true: If points D, E, F are chosen on BC, AC, AB respectively so that
then D, E, F are collinear. The converse is often included as part of the theorem. (Note that the converse of the weaker, unsigned statement is not necessarily true.) The theorem is very similar to Ceva's theorem in that their equations differ only in sign. By re writing each in terms of cross ratios, the two theorems may be seen as projective duals.
Эквивалентно,
In Euclidean geometry, Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle △ABC, and a transversal line that crosses BC, AC, AB at points D, E, F respectively, with D, E, F distinct from A, B, C. A weak version of the theorem states that
where "| |" denotes absolute value (i. e., all segment lengths are positive). The theorem can be strengthened to a statement about signed lengths of segments, which provides some additional information about the relative order of collinear points. Here, the length is taken to be positive or negative according to whether A is to the left or right of B in some fixed orientation of the line; for example, is defined as having positive value when F is between A and B and negative otherwise. The signed version of Menelaus's theorem states
Equivalently,
Some authors organize the factors differently and obtain the seemingly different relation
but as each of these factors is the negative of the corresponding factor above, the relation is seen to be the same. The converse is also true: If points D, E, F are chosen on BC, AC, AB respectively so that
then D, E, F are collinear. The converse is often included as part of the theorem. (Note that the converse of the weaker, unsigned statement is not necessarily true.) The theorem is very similar to Ceva's theorem in that their equations differ only in sign. By re writing each in terms of cross ratios, the two theorems may be seen as projective duals.
Некоторые авторы располагают множители по-другому и получают кажущееся иным соотношение:
In Euclidean geometry, Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle △ABC, and a transversal line that crosses BC, AC, AB at points D, E, F respectively, with D, E, F distinct from A, B, C. A weak version of the theorem states that
where "| |" denotes absolute value (i. e., all segment lengths are positive). The theorem can be strengthened to a statement about signed lengths of segments, which provides some additional information about the relative order of collinear points. Here, the length is taken to be positive or negative according to whether A is to the left or right of B in some fixed orientation of the line; for example, is defined as having positive value when F is between A and B and negative otherwise. The signed version of Menelaus's theorem states
Equivalently,
Some authors organize the factors differently and obtain the seemingly different relation
but as each of these factors is the negative of the corresponding factor above, the relation is seen to be the same. The converse is also true: If points D, E, F are chosen on BC, AC, AB respectively so that
then D, E, F are collinear. The converse is often included as part of the theorem. (Note that the converse of the weaker, unsigned statement is not necessarily true.) The theorem is very similar to Ceva's theorem in that their equations differ only in sign. By re writing each in terms of cross ratios, the two theorems may be seen as projective duals.
но поскольку каждый из этих множителей является отрицательным по отношению к соответствующему множителю выше, соотношение остается тем же самым. Обратное также верно: если точки D, E и F выбраны на сторонах BC, AC и AB соответственно так, что
In Euclidean geometry, Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle △ABC, and a transversal line that crosses BC, AC, AB at points D, E, F respectively, with D, E, F distinct from A, B, C. A weak version of the theorem states that
where "| |" denotes absolute value (i. e., all segment lengths are positive). The theorem can be strengthened to a statement about signed lengths of segments, which provides some additional information about the relative order of collinear points. Here, the length is taken to be positive or negative according to whether A is to the left or right of B in some fixed orientation of the line; for example, is defined as having positive value when F is between A and B and negative otherwise. The signed version of Menelaus's theorem states
Equivalently,
Some authors organize the factors differently and obtain the seemingly different relation
but as each of these factors is the negative of the corresponding factor above, the relation is seen to be the same. The converse is also true: If points D, E, F are chosen on BC, AC, AB respectively so that
then D, E, F are collinear. The converse is often included as part of the theorem. (Note that the converse of the weaker, unsigned statement is not necessarily true.) The theorem is very similar to Ceva's theorem in that their equations differ only in sign. By re writing each in terms of cross ratios, the two theorems may be seen as projective duals.
то точки D, E и F коллинеарны. Обратное часто включается в формулировку теоремы. (Следует отметить, что обратное к более слабой, незазнаковой формулировке не обязательно верно.) Теорема очень похожа на теорему Чевы тем, что их уравнения различаются только знаком. Переписывая каждую из них в терминах двойных отношений, можно увидеть, что обе теоремы являются проективными дуалами.
In Euclidean geometry, Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle △ABC, and a transversal line that crosses BC, AC, AB at points D, E, F respectively, with D, E, F distinct from A, B, C. A weak version of the theorem states that
where "| |" denotes absolute value (i. e., all segment lengths are positive). The theorem can be strengthened to a statement about signed lengths of segments, which provides some additional information about the relative order of collinear points. Here, the length is taken to be positive or negative according to whether A is to the left or right of B in some fixed orientation of the line; for example, is defined as having positive value when F is between A and B and negative otherwise. The signed version of Menelaus's theorem states
Equivalently,
Some authors organize the factors differently and obtain the seemingly different relation
but as each of these factors is the negative of the corresponding factor above, the relation is seen to be the same. The converse is also true: If points D, E, F are chosen on BC, AC, AB respectively so that
then D, E, F are collinear. The converse is often included as part of the theorem. (Note that the converse of the weaker, unsigned statement is not necessarily true.) The theorem is very similar to Ceva's theorem in that their equations differ only in sign. By re writing each in terms of cross ratios, the two theorems may be seen as projective duals.
Доказательство с использованием гомотеций
В следующем доказательстве используются только понятия аффинной геометрии, в частности гомотетии. Независимо от того, лежат ли точки D, E, F на одной прямой, существуют три гомотетии с центрами D, E, F, которые соответственно отображают точку B в точку C, точку C в точку A и точку A в точку B. Композиция этих трех гомотетий является элементом группы гомотетических преобразований, фиксирующим точку B, то есть это гомотетия с центром B, возможно, с коэффициентом 1 (в этом случае это тождественное преобразование). Эта композиция фиксирует прямую DE тогда и только тогда, когда точка F лежит на прямой DE (поскольку первые две гомотетии, безусловно, фиксируют прямую DE, а третья фиксирует её только в том случае, если F лежит на DE). Следовательно, точки D, E, F лежат на одной прямой тогда и только тогда, когда эта композиция является тождественным преобразованием, что означает, что модуль произведения трех коэффициентов равен 1:
что эквивалентно данному уравнению.
История
Неизвестно, кто на самом деле открыл эту теорему; однако, самое раннее из сохранившихся изложений встречается в "Сферике" Менелая. В этой книге плоская версия теоремы используется как лемма для доказательства сферической версии. В "Альмагесте" Птолемей применяет теорему при решении ряда задач сферической астрономии. В период исламского Золотого века мусульманские ученые посвятили ряд работ изучению теоремы Менелая, которую они называли «предложением о секущих» (шакл аль-катта). Полный четырехугольник в их терминологии именовался «фигурой секущих». Или это были работы, составленные в виде самостоятельных трактатов, такие как:
«Трактат о фигуре секущих» (Risala fi shakl al qatta'') Табита ибн Курры. «Открытие завесы над тайнами фигуры секущих» (Kashf al qina' 'an asrar al shakl al qatta'), также известная как «Книга о фигуре секущих» (Kitab al shakl al qatta') или в Европе как «Трактат о полном четырехугольнике». О потерянном трактате упоминали Шараф ад-Дин ат-Туси и Насир ад-Дин ат-Туси. Работы Аль Сиджи. «Тахдиб» Абу Насра ибн Ирака. Рошди Рашид и Афанасий Пападопулос, «Сферика Менелая: ранний перевод и версия аль-Махани/аль-Харави» (Критическое издание «Сферики Менелая» по арабским рукописям, с историческими и математическими комментариями), De Gruyter, Серия: Scientia Graeco Arabica, 21, 2017, 890 страниц.