مبرهنة منلاوس

مبرهنة مينلاوس، الحالة الأولى: يمر المستقيم DEF داخل المثلث ABC.

في الهندسة الرياضية،مبرهنة مينلاوس إنگليزية: Menelaus' theorem هي مبرهنة تم صياغتها من قبل مينلاوس الإسكندري وهي تتعلق بالمثلثات في المستوي.

إذا كانت النقاط الثلاثة A، B، C تشكل مثلثاً هو ABC، وكانت النقاط D، E، F تقع على المستقيمات BC، AC، AB عندها تنص المبرهنة على أن النقاط الثلاثة D, E, F تقع على مستقيم واحد فقط وفقط إذا تحققت العلاقة:

${\frac {AF}{FB}}\cdot {\frac {BD}{DC}}\cdot {\frac {CE}{EA}}=-1$

حيث يسمح في هذه النظرية لأطوال الأضلاع بأخذ قيمة سالبة. مثلاً تأخذ النسبة AF / FB قيمة موجبة فقط إذا قطع المستقيم DEF الضلع AB، وبشكل مماثل للكسور الأخرى.

Equivalently,

$${\overline {AF}}\times {\overline {BD}}\times {\overline {CE}}=-{\overline {FB}}\times {\overline {DC}}\times {\overline {EA}}.$$

Some authors organize the factors differently and obtain the seemingly different relation

$${\frac {\overline {FA}}{\overline {FB}}}\times {\frac {\overline {DB}}{\overline {DC}}}\times {\frac {\overline {EC}}{\overline {EA}}}=1,$$

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

$${\frac {\overline {AF}}{\overline {FB}}}\times {\frac {\overline {BD}}{\overline {DC}}}\times {\frac {\overline {CE}}{\overline {EA}}}=-1,$$

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.

 البراهين

مبرهنة مينلاوس، الحالة الثانية: المستقيم DEF هو بالكامل خارج المثلث △ABC.

 برهان قياسي

A proof given by John Wellesley Russell uses Pasch's axiom to consider cases where a line does or does not meet a triangle. First, the sign of the left-hand side will be negative since either all three of the ratios are negative, the case where the line DEF misses the triangle (see diagram), or one is negative and the other two are positive, the case where DEF crosses two sides of the triangle.

To check the magnitude, construct perpendiculars from A, B, C to the line DEF and let their lengths be a, b, c respectively. Then by similar triangles it follows that

$$\left|{\frac {\overline {AF}}{\overline {FB}}}\right|=\left|{\frac {a}{b}}\right|,\quad \left|{\frac {\overline {BD}}{\overline {DC}}}\right|=\left|{\frac {b}{c}}\right|,\quad \left|{\frac {\overline {CE}}{\overline {EA}}}\right|=\left|{\frac {c}{a}}\right|.$$

Therefore,

$$\left|{\frac {\overline {AF}}{\overline {FB}}}\right|\times \left|{\frac {\overline {BD}}{\overline {DC}}}\right|\times \left|{\frac {\overline {CE}}{\overline {EA}}}\right|=\left|{\frac {a}{b}}\times {\frac {b}{c}}\times {\frac {c}{a}}\right|=1.$$

For a simpler, if less symmetrical way to check the magnitude, draw CK parallel to AB where DEF meets CK at K. Then by similar triangles

$$\left|{\frac {\overline {BD}}{\overline {DC}}}\right|=\left|{\frac {\overline {BF}}{\overline {CK}}}\right|,\quad \left|{\frac {\overline {AE}}{\overline {EC}}}\right|=\left|{\frac {\overline {AF}}{\overline {CK}}}\right|,$$

and the result follows by eliminating CK from these equations.

The converse follows as a corollary. Let D, E, F be given on the lines BC, AC, AB so that the equation holds. Let F' be the point where DE crosses AB. Then by the theorem, the equation also holds for D, E, F'. Comparing the two,

$${\frac {\overline {AF}}{\overline {FB}}}={\frac {\overline {AF'}}{\overline {F'B}}}\ .$$

But at most one point can cut a segment in a given ratio so F = F'.

 A proof using homotheties

Homothetie centers D, E F are colinear iff the composition is identity.

The following proof uses only notions of affine geometry, notably homotheties. Whether or not D, E, F are collinear, there are three homotheties with centers D, E, F that respectively send B to C, C to A, and A to B. The composition of the three then is an element of the group of homothety-translations that fixes B, so it is a homothety with center B, possibly with ratio 1 (in which case it is the identity). This composition fixes the line DE if and only if F is collinear with D, E (since the first two homotheties certainly fix DE, and the third does so only if F lies on DE). Therefore D, E, F are collinear if and only if this composition is the identity, which means that the magnitude of the product of the three ratios is 1:

$${\frac {\overrightarrow {DC}}{\overrightarrow {DB}}}\times {\frac {\overrightarrow {EA}}{\overrightarrow {EC}}}\times {\frac {\overrightarrow {FB}}{\overrightarrow {FA}}}=1,$$

which is equivalent to the given equation.

 التاريخ

It is uncertain who actually discovered the theorem; however, the oldest extant exposition appears in Spherics by Menelaus. In this book, the plane version of the theorem is used as a lemma to prove a spherical version of the theorem.

In Almagest, Ptolemy applies the theorem on a number of problems in spherical astronomy. During the Islamic Golden Age, Muslim scholars devoted a number of works that engaged in the study of Menelaus's theorem, which they referred to as "the proposition on the secants" (shakl al-qatta'). The complete quadrilateral was called the "figure of secants" in their terminology. Al-Biruni's work, The Keys of Astronomy, lists a number of those works, which can be classified into studies as part of commentaries on Ptolemy's Almagest as in the works of al-Nayrizi and al-Khazin where each demonstrated particular cases of Menelaus's theorem that led to the sine rule, or works composed as independent treatises such as:

• The "Treatise on the Figure of Secants" (Risala fi shakl al-qatta') by Thabit ibn Qurra.
• Husam al-Din al-Salar's Removing the Veil from the Mysteries of the Figure of Secants (Kashf al-qina' 'an asrar al-shakl al-qatta'), also known as "The Book on the Figure of Secants" (Kitab al-shakl al-qatta') or in Europe as The Treatise on the Complete Quadrilateral. The lost treatise was referred to by Sharaf al-Din al-Tusi and Nasir al-Din al-Tusi.
• Work by al-Sijzi.
• Tahdhib by Abu Nasr ibn Iraq.
• Roshdi Rashed and Athanase Papadopoulos, Menelaus' Spherics: Early Translation and al-Mahani'/al-Harawi's version (Critical edition of Menelaus' Spherics from the Arabic manuscripts, with historical and mathematical commentaries), De Gruyter, Series: Scientia Graeco-Arabica, 21, 2017, 890 pages. ISBN 978-3-11-057142-4

In the West the theorem was rediscovered by the Italian mathematician Giovanni Ceva.

 انظر أيضاً

• مبرهنة سيفا

 وصلات خارجية

• Alternate proof of Menelaus's theorem, from PlanetMath
• Menelaus From Ceva
• Ceva and Menelaus Meet on the Roads
• Menelaus and Ceva at MathPages
• Demo of Menelaus's theorem by Jay Warendorff. The Wolfram Demonstrations Project.
• Eric W. Weisstein, Menelaus' Theorem at MathWorld.