Triangle

Triangle#

A triangle is a geometric figure with three sides. This page explores the various properties of triangles.

Similar triangles#

Given two triangles \(ABC\) and \(DEF\).

A B C D E F

Triangles are called similar (\(\Delta ABC \sim \Delta EDF\)) if the following holds:

\[\frac{AB}{DE} = \frac{AC}{DF} = \frac{BC}{EF}\]

and

\[\angle A = \angle D, \quad \angle B = \angle E, \quad \angle C = \angle F.\]

The value

\[k = \frac{P_{ABC}}{P_{DEF}}\]

is called the coefficient of similarity, where:

  • \(P_{ABC}\) is the perimeter of triangle \(ABC\).

  • \(P_{DEF}\) is the perimeter of triangle \(DEF\).

For coefficient of similarity is a fair expression:

\[k^2=\frac{S_{ABC}}{S_{DEF}}\]

Where:

  • \(S_{ABC}\) is the area of triangle \(ABC\).

  • \(S_{DEF}\) is the area of triangle \(DEF\).

Triangle area#

There are a few methods for calculating the area of the triangle (\(S\)).

\[S = \frac{1}{2}ah_a\]

Where \(a\) is the length of the triangle and \(h_a\) is a heights its heights.

\[S = \frac{1}{2} bc \sin{\alpha}\]

Here, \(b\) and \(c\) are sides of a triangle, and \(\alpha\) is the angle between them.

Heron’s formula:

\[S = \sqrt{p(p-a)(p-b)(p-c)}\]

If \(a, b, c\) are the sides of traingle and \(p\) is half of the perimeter.

Median#

Median is a line segment that connects a vertex to the midpoint of the opposite side.

The following picture illustrates the median \(AO\) of the \(\angle BAC\).

A B C O m a b c γ

In the picture, \(AC\) is denoted as \(M\) for the convenience of the following mathematical descriptions.

Distance#

To calculate the distance of the median use formulas:

  • \(m=\frac{1}{2}\sqrt{2a^2+2b^2-c^2}\).

  • \(m=\frac{1}{2}\sqrt{2a^2+2b^2+2ab\,\cos(\gamma)}.\)

To prove the identity \(M=\frac{1}{2}\sqrt{2a^2+2b^2-c^2}\), extend the \(AO\) line with the \(OD\) line, which length equals to \(m\), and draw the \(BD\) and \(CD\) lines. As shown in the following picture:

A B C O m a b c D

The shape \(ABDC\) is a parallelogram because it is a four-cornered figure whose diagonals are divided into equal segments at their intersection.

Using properties of the diagonals of the parallelogramm:

\[c^2 + (2m)^2 = 2a^2 + 2b^2\]

We can obtain the following by using simple transformations of the formula:

\[m = \frac{1}{2}\sqrt{2a^2 + 2b^2 - c^2}\]