Triangle

A triangle is a geometric figure that is made up of three vertices that are connected by three lines.

Description

Look at the following picture: The picture shows a triangle which is formed by the vertices A, B and C; so it is a triangle ABC. Here we find three sides: AB, BC, AC. Also note that these sides are additionally named with lower case letters. This naming has a rule - opposite the vertex A we have the side a. Opposite the vertex B is the page b, and opposite the vertex C is the page c. Thus, the opposite side is always named after the vertex; the side not formed by the vertex.

Each triangle has three interior angles, which we usually denote by the Greek letters alpha α, beta β, and gamma γ. The sum of all three interior angles must always give 180 degrees.

A triangle has no diagonals, but it does have lines of gravity and height.

Triangle inequality

The triangle inequality is an important relationship that holds in a triangle. It holds that the sum of the lengths of any two sides is always greater than the length of the third, remaining side. Writing this down:

$$\begin{eqnarray} |a|+|b|&>&|c|\\ |a|+|c|&>&|b|\\ |b|+|c|&>&|a| \end{eqnarray}$$

What would happen if this inequality did not hold? That is, if it were true that one side is longer than the sum of the remaining two? A triangle could not be formed because the two sides would be too short and would not "reach" each other.

If there was equality, i.e. two sides were as long in the sum as the third, side, then when trying to draw a triangle, all the points would lie on the same line:

The triangle inequality is also used in definitions of other terms, often related to distance in some way, where this principle is most natural. If we are talking about distances in the real world, it is certainly true that a direct journey (as the crow flies) from Prague to Brno is certainly shorter than a journey from Prague to Liberec and then from Liberec to Brno.

Types of triangles

There are several different types of triangles, depending on the length of the sides and the angles.

• Anequilateral triangle has all sides the same length. At the same time, all interior angles have a magnitude of 60 degrees.
• Anisosceles triangle has two sides of equal length and the third side has a different length. The sides that are equal in length are called the arms, and the third side is called the base. The angle that the arms of such a triangle make with the base is always the same.
• Anacute triangle has every interior angle acute, i.e., less than 90 degrees.
• A right-angled triangle has just one right angle, i.e., of 90 degrees. A triangle cannot have two right angles because the sum of the interior angles is equal to 180 - the third angle would then have to be of size zero, which is not possible. In a right triangle, the famous Pythagorean theorem holds. For more information, read the separate article on right triangles.
• Anobtuse triangle has just one angle, its size is greater than 90 degrees. Again, note that a triangle cannot have two angles greater than 90 degrees, so the remaining two angles must necessarily be acute, i.e., less than 90 degrees.