Triangle (2d)

A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry.

Types of triangles:

Scalene, isosceles, equilateral, right-angled, acute, obtuse. Each has its own unique characteristics.

Key elements of a triangle:

Sides, angles, vertices, but also medians, altitudes, angle bisectors, and perpendicular bisectors. These lines and segments reveal hidden connections within the triangle.

Fundamental theorems:

The Pythagorean theorem, the Triangle Inequality Theorem, and the Angle Sum Theorem are cornerstones.

Special points of a triangle:

The centroid, circumcenter, incenter, and orthocenter – each with its own unique definition and significance. Did you know they sometimes coincide and sometimes form interesting lines like the Euler line?

Trigonometry and triangles:

The relationships between the angles and sides of triangles are beautifully captured by trigonometric functions like sine, cosine, and tangent.

Area formulas:

Heron's formula, 1/2 ×base×height, and using trigonometry – there are multiple ways to calculate the area of a triangle.

Congruence and similarity:

Understanding when two triangles are identical or have the same shape but different sizes is crucial for many geometric proofs and applications.

Geometric constructions:

Using only a compass and straightedge to create specific triangles and their elements is a classic and elegant part of geometry.