A triangle is a polygon with three sides and three vertices, or corners. It is one of the basic shapes in geometry and is the simplest polygon. In this example, we have a triangle ABC with three sides: AB, BC, and CA, and three vertices: A, B, and C.
Triangles can be classified in different ways. Based on their sides, triangles can be equilateral, where all three sides are equal; isosceles, where two sides are equal; or scalene, where all sides have different lengths. Based on their angles, triangles can be acute, where all angles are less than 90 degrees; right, where one angle is exactly 90 degrees; or obtuse, where one angle is greater than 90 degrees.
Triangles have several important properties. First, the sum of the interior angles in any triangle is always 180 degrees. This means that if we know two angles, we can easily calculate the third. Second, the sum of the lengths of any two sides must be greater than the length of the third side. This is known as the triangle inequality. Third, the area of a triangle can be calculated using the formula: one-half times the base times the height.
Triangles have several special lines and points. A median is a line that connects a vertex to the midpoint of the opposite side. Each triangle has three medians, and they all intersect at a single point called the centroid, which is the center of mass of the triangle. An altitude is a perpendicular line from a vertex to the opposite side. The three altitudes of a triangle intersect at a point called the orthocenter. Other important points include the circumcenter, which is equidistant from all vertices, and the incenter, which is equidistant from all sides.
To summarize what we've learned about triangles: A triangle is a polygon with three sides and three vertices. Triangles can be classified by their sides as equilateral, isosceles, or scalene, and by their angles as acute, right, or obtuse. The sum of interior angles in any triangle is always 180 degrees. The area of a triangle can be calculated as one-half times the base times the height. Triangles have special points like the centroid, orthocenter, and circumcenter, which have important geometric properties. Understanding triangles is fundamental to geometry and has applications in architecture, engineering, and many other fields.