The Triangle

A triangle is one of the simplest yet most important shapes in geometry. It has exactly three sides, three vertices, and three angles that always add up to $180°$. Every triangle also has four classical centers: the centroid, circumcenter, incenter, and orthocenter.

Keywords: triangle, polygon, vertices, angles, sides, centroid, circumcenter, incenter, orthocenter, triangle centers, geometry basics

Difficulty: beginner

About Triangles

A triangle is the simplest polygon — a closed shape made of straight lines. It has exactly:

• 3 sides (line segments)
• 3 vertices (corner points)
• 3 interior angles

The Magic Number: 180°

Here's something amazing: no matter what shape your triangle takes — tall and skinny, short and wide, or perfectly balanced — the three interior angles always add up to exactly $180°$.

\angle A + \angle B + \angle C = 180°
    

Try dragging the vertices around and watch how the angles change, but their sum stays the same!

Why Triangles Matter

Triangles are everywhere in the real world:

• Bridges and buildings use triangular supports because triangles are incredibly strong — they can't be "squished" like rectangles can.
• Pizza slices and sandwiches cut diagonally are triangles.
• Mountains often appear triangular against the sky.
• Traffic signs like yield signs are triangles.

Engineers love triangles because they're the only polygon that's rigid — you can't change a triangle's shape without changing the length of at least one side.

Types of Triangles

Triangles can be classified by their sides:

• Equilateral: All three sides are equal (and all angles are $60°$)
• Isosceles: Two sides are equal
• Scalene: All three sides are different

Or by their angles:

• Acute: All angles are less than $90°$
• Right: One angle is exactly $90°$
• Obtuse: One angle is greater than $90°$

Special Points

Every triangle has special points called centers:

• Centroid (G): Where the three medians meet — the triangle's "balance point"
• Circumcenter (O): Where the perpendicular bisectors meet — center of the circle through all vertices
• Incenter (I): Where the angle bisectors meet — center of the inscribed circle
• Orthocenter (H): Where the altitudes meet

These four points have a surprising relationship — three of them always lie on a single line called the Euler line!