Equilateral Triangle: Definition and Examples

An equilateral triangle is one of the best-known shapes in geometry. Why is that? Because it has a very simple yet very important structure. In this kind of triangle, all sides are equal. In addition, all of its interior angles are equal as well.

If each side of a triangle has the same length, then it is an equilateral triangle. For example, if all three sides are \(10\) cm, the triangle is equilateral. Because of this, it can also be viewed as a special case of a regular polygon.

In this article, we will look at the definition of an equilateral triangle, its main properties, and typical examples of problems and practical questions.

Equilateral Triangle: Definition and Main Properties

As already mentioned, an equilateral triangle is a triangle in which all sides have the same length. However, its special features do not end there. Since the sides are equal, the interior angles are equal as well. Each of them measures \(60^\circ\).

Take a look at the following figure.

In the figure, the sides of triangle \(ABC\) are equal, that is,

\[
AB = BC = AC.
\]

All interior angles are equal too:

\[
\angle A = \angle B = \angle C = 60^\circ.
\]

So, by definition, triangle \(ABC\) is equilateral.

Properties of an Equilateral Triangle

An equilateral triangle has several important properties. These properties help us not only recognize this figure, but also solve geometry problems.

In an equilateral triangle, all sides are equal.
All interior angles of an equilateral triangle are \(60^\circ\).
An equilateral triangle is a regular polygon with three sides.
A perpendicular drawn from a vertex to the opposite side divides that side into two equal parts. At the same time, the angle at the vertex is also divided into two equal angles of \(30^\circ\).

The centers of the inscribed and circumscribed circles in an equilateral triangle coincide at a single point.
The center of an equilateral triangle divides the height in the ratio \(2:1\), counting from the vertex. That is,

\[
BO = 2 \cdot OH.
\]

In an equilateral triangle, the median, the angle bisector, and the altitude drawn from the same vertex all coincide.
All of these segments are also axes of symmetry of the triangle.

So, an equilateral triangle is not just a triangle with equal sides. It is a figure with a very orderly and symmetrical structure.

Equilateral Triangle: Examples of Problems and Practical Questions

Now let us move on to examples. They help us better understand the definition and properties of an equilateral triangle in practice.

Example 1. What is an equilateral triangle in geometry?

An equilateral triangle is a triangle in which all sides are equal. Since all of its angles are also equal and each of them is \(60^\circ\), such a triangle is also called equiangular.

Example 2. Under what conditions can we say that a triangle is equilateral?

A triangle can be considered equilateral if one of the following conditions holds:

Two of its angles are \(60^\circ\).
Two sides are equal, and one of the angles is \(60^\circ\).
All medians of the triangle are equal.
All angle bisectors of the triangle are equal.
All altitudes of the triangle are equal.

These conditions are related to the properties of an equilateral triangle and help us recognize it in geometry problems.

Example 3. What is the sum of all angles in an equilateral triangle?

An equilateral triangle has three interior angles. Since each of them is \(60^\circ\), the sum of all angles is

\[
60^\circ + 60^\circ + 60^\circ = 180^\circ.
\]

So, just like in any other triangle, the sum of the interior angles of an equilateral triangle is \(180^\circ\).

Example 4. Find the angle bisector of equilateral triangle \(ABC\) with side \(4\) cm

According to the given information, \(ABC\) is an equilateral triangle. In such a triangle, the angle bisector, median, and altitude drawn from the same vertex coincide.

Let us draw the angle bisector from vertex \(A\) to side \(BC\). Then we get a right triangle \(ABK\), in which

\[
BK = \frac{4}{2} = 2,\qquad AB = 4.
\]

To find \(AK\), we use the Pythagorean theorem:

\[
\begin{gathered}
AB^2 = AK^2 + BK^2,
\\[6pt]
AK^2 = AB^2 – BK^2,
\\[6pt]
AK = \sqrt{AB^2 – BK^2} = \sqrt{4^2 – 2^2} = \sqrt{16 – 4} = \sqrt{12} \approx 3.464.
\end{gathered}
\]

Thus, the angle bisector of the equilateral triangle is \(3.464\) cm.

Example 5. Find the radius \(OK\) of the inscribed circle in equilateral triangle \(ABC\) if the height of the triangle is \(15\) cm

As already mentioned, the center of the inscribed circle in an equilateral triangle coincides with the point where the altitudes, medians, and angle bisectors intersect. It is also important to remember that the height is divided by this point in the ratio \(2:1\), counting from the vertex.

So, the radius of the inscribed circle is one third of the height:

\[
OK = \frac{AK}{3} = \frac{15}{3} = 5.
\]

Therefore, the radius of the circle inscribed in the equilateral triangle is \(5\) cm.

From a Flowchart to Code: Try Programming It Yourself

Now look at the flowchart below not only as an illustration, but also as the foundation for a small educational project. Why not try implementing it in your favorite programming language? This kind of work helps you see how a geometric idea turns into a clear algorithm, and then into program code that you can run, test, and expand with your own examples. It is not only interesting, but also useful, because it will help you better understand both the equilateral triangle itself and the practical use of mathematics in programming.

Equilateral Triangle: Definition, Properties, and Examples