A triangle is a polygon with three edges and three vertices, or corners. It is one of the basic shapes in geometry. As you can see, this triangle has three vertices labeled A, B, and C, and three edges connecting these vertices. Triangles are fundamental building blocks in many geometric constructions and have numerous important properties.
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 is shown by the angles alpha, beta, and gamma. The area of a triangle can be calculated as one-half times the base times the height. The perimeter is simply the sum of all three sides. Another fundamental property is the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side. This is why not all combinations of three lengths can form a triangle.
Triangles have several special lines and points. Medians are lines that connect a vertex to the midpoint of the opposite side. All three medians intersect at a single point called the centroid, which is the center of mass of the triangle. Altitudes are perpendicular lines drawn from a vertex to the opposite side. Angle bisectors divide each angle into two equal parts. These special lines and points have important properties and applications in geometry, physics, and engineering.
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; or 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, while the perimeter is the sum of all sides. Triangles have special lines such as medians, altitudes, and angle bisectors, which intersect at special points like the centroid, orthocenter, and incenter. These properties make triangles fundamental shapes in geometry with numerous applications.