Menelaus' Theorem

Menelaus' theorem is a powerful tool in geometry that helps us solve problems involving collinear points. It is named after Menelaus of Alexandria, a Greek mathematician who lived in the first century AD.

Statement of the Theorem

The statement of Menelaus' theorem is as follows:

Given a triangle $ABC$ and a line $l$ that intersects its sides $AB$, $BC$, and $CA$ at points $D$, $E$, and $F$ respectively, the following equation holds:

$\frac{AD}{DB} \cdot \frac{BE}{EC} \cdot \frac{CF}{FA} = 1$

Proof

To prove Menelaus' theorem, we can use the following strategy:

1. Use similar triangles to express the ratios $\frac{AD}{DB}$, $\frac{BE}{EC}$, and $\frac{CF}{FA}$ in terms of other ratios.
2. Simplify the resulting expression to show that it is equal to $1$.

Let's see how this works in practice. Consider the following diagram:

We want to prove that $\frac{AD}{DB} \cdot \frac{BE}{EC} \cdot \frac{CF}{FA} = 1$. To do this, we can use similar triangles. Specifically, we can use the fact that triangles $ADC$ and $BDC$ are similar, triangles $BEC$ and $AEC$ are similar, and triangles $AFB$ and $CFB$ are similar.

Using these similar triangles, we can express the ratios $\frac{AD}{DB}$, $\frac{BE}{EC}$, and $\frac{CF}{FA}$ in terms of other ratios:

$\frac{AD}{DB} = \frac{AC}{BC} \cdot \frac{\sin \angle ACD}{\sin \angle BCD}$

$\frac{BE}{EC} = \frac{AB}{AC} \cdot \frac{\sin \angle ABE}{\sin \angle CBE}$

$\frac{CF}{FA} = \frac{BC}{AB} \cdot \frac{\sin \angle BCF}{\sin \angle ACF}$

Substituting these expressions into the original equation, we obtain:

$\frac{AC}{BC} \cdot \frac{\sin \angle ACD}{\sin \angle BCD} \cdot \frac{AB}{AC} \cdot \frac{\sin \angle ABE}{\sin \angle CBE} \cdot \frac{BC}{AB} \cdot \frac{\sin \angle BCF}{\sin \angle ACF} = 1$

Simplifying this expression, we get:

$\frac{\sin \angle ACD}{\sin \angle BCD} \cdot \frac{\sin \angle ABE}{\sin \angle CBE} \cdot \frac{\sin \angle BCF}{\sin \angle ACF} = 1$

This is a well-known identity in trigonometry, known as the trigonometric form of Ceva's theorem. Therefore, we have proven Menelaus' theorem.

Applications

Menelaus' theorem has many applications in geometry. Here are a few examples:

• It can be used to prove that the medians of a triangle are concurrent.
• It can be used to prove that the altitudes of a triangle are concurrent.
• It can be used to prove that the angle bisectors of a triangle are concurrent.
• It can be used to solve various problems involving collinear points in triangles, such as the famous "butterfly theorem".

Conclusion

Menelaus' theorem is a powerful tool in geometry that helps us solve problems involving collinear points. Its statement may seem daunting at first, but its proof is based on simple principles of similarity and trigonometry. With its many applications, Menelaus' theorem is a valuable tool for any student of geometry to have in their toolbox.

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メネラウスの定理[JA]