# Triangle

A **triangle** is a three-sided polygon defined by its three angles.

## Overview

As one of the basic shapes in Euclidean geometry, the triangle consists of a set of points bounded by a closed polyline.

The triangle's corners, known as **vertices**, are defined by three distinct points (A, B, C) that are not in a straight line.

A vertex is referred to as the **vertex opposite to a side** when it is not part of that side.

Each pair of sides in a triangle defines an **internal angle**, which is formed at a vertex of the triangle and flanked by the two sides.

An angle is **adjacent to a side** if its vertex matches one of the side's endpoints, and one of its arms extends along the side itself.

For instance, angle β is adjacent to sides AB and BC alike.

Thus, two angles are adjacent to each side of the triangle.

Angles α and β, for example, flank side AB.

An angle is termed an **opposite angle** to a side if it does not adjoin the side.

For example, angle γ stands opposite side AB.

No matter the triangle's shape, **the sum of its internal angles always equals 180°**.

$$ \alpha + \beta + \gamma = 180° $$

Corresponding external angles are associated with each internal angle of the triangle.

Derived from the Greek words for "three" and "angle," the triangle is a versatile shape used across various fields, such as physics and mathematics. It plays a crucial role in trigonometry, which studies the relationships between a triangle's sides and angles.

## Triangle Types

Triangles are versatile shapes that can be classified in multiple ways, each revealing unique properties and characteristics.

By Sides:

**Equilateral Triangle**

An equilateral triangle features three sides of identical length, resulting in all three internal angles being equal to 60 degrees.

**Isosceles Triangle**

Characterized by two sides of the same length, an isosceles triangle also boasts two equal angles adjacent to its base.

**Scalene Triangle**

With three unequal sides, each angle in a scalene triangle is also distinct, adding to its asymmetrical beauty.

By Angles:

**Acute Triangle**

An acute triangle has all its angles less than 90 degrees, showcasing a sharp, slender appearance.

**Obtuse Triangle**

Featuring a single angle greater than 90 degrees, the obtuse triangle offers a broad, expansive feel.

**Right Triangle**

The classic right triangle is easily recognized by its singular 90-degree angle, forming the basis for trigonometry.

Unique to the right triangle, the sides that form the 90° angle are referred to as "legs", with the side opposite the right angle known as the "hypotenuse".

## Triangle Formulas

Key formulas to measure various aspects of a triangle include:

**Area**

To find a triangle's area (A), multiply the base (b) by the height (h) and divide by two: $$ A = \frac{\text{b} \cdot \text{h}}{2} $$.Alternatively, if I only know the lengths of the sides, I can use Heron's formula. $$ A = \sqrt{p(p-l_1)(p-l_2)(p-l_2)} $$ Where \( l_1 \), \( l_2 \), and \( l_3 \) are the sides and \(p = \frac{l_1+l_2+l_3}{2}\) is the semiperimeter of the triangle.

**Perimeter**

The perimeter (P) is the total length around the triangle, calculated by adding the lengths of its three sides: $$ P = a + b + c $$, where a, b, and c are the side lengths.

Isosceles and equilateral triangles have their own unique formulas for calculating area and perimeter.

## Triangle Centers

Triangles are known for having several important centers, including the centroid, circumcenter, and incenter, among others.

**Orthocenter**

The point where a triangle's three altitudes intersect is called the orthocenter.**Note:**Triangles have three altitudes. Depending on the triangle type, the orthocenter can be inside the triangle, outside, or at one of the vertices. It lies outside in an obtuse scalene triangle, inside in an acute scalene triangle, and at the right angle vertex in a right scalene triangle.**Centroid**

The centroid is the point where a triangle's three medians intersect, located inside the triangle.

**Incenter**

The incenter is the intersection point of the triangle's three angle bisectors and is the center of the inscribed circle.

**Circumcenter**

The circumcenter, where the triangle's three perpendicular bisectors meet, is the center of the circumscribed circle.

The orthocenter, centroid, circumcenter, and incenter of a triangle are all concyclic, residing on the Euler circle.

## Triangle Properties

Let's delve into some properties of triangles in geometry.

**All triangles have interior angles adding up to 180°**

This means the angles inside any triangle always total a straight angle. This fundamental rule applies across all triangle types. $$ \alpha + \beta + \gamma = 180° $$**The sum of any two internal angles is less than 180°**

Specifically, in any triangle, combining any two internal angles will always give you a sum less than a straight angle (180°).

**At least two angles in any triangle are acute**

Meaning, it's impossible for a triangle to have more than one right angle or one obtuse angle. This ensures the interior angles don't exceed 180°, aligning with the principle that the sum of any two internal angles is less than a straight angle.

Contradicting this would mean defying the theorem regarding the sum of any two internal angles being less than 180°.

**Exterior Angle Theorem**

Any exterior angle of a triangle (β_{e}) surpasses each of its non-adjacent interior angles (α and γ), showcasing the unique relationship between a triangle's angles.

**Pythagorean Theorem**

A cornerstone of geometry, this theorem reveals that in a right-angled triangle, the hypotenuse's square (the side opposite the right angle) equals the sum of the squares on the other two sides.- Every triangle features a singular centroid, the point of convergence for its medians (the segments connecting each vertex with the midpoint of the opposing side), serving as the triangle's balance point.

And that's just the tip of the iceberg when it comes to triangle properties.