Geometric Relationships in Triangles

This page focuses on key formulas and relationships within triangles, a fundamental shape in planimetry. It covers area calculations, trigonometric ratios, and special lines within triangles.

The page begins with several formulas for calculating a triangle's area:

Definition: The area of a triangle can be calculated using the formula A = 1/2 * base * height, where the height is perpendicular to the base.

Additional area formulas are provided using different variables, including one using the semiperimeter and side lengths (Heron's formula).

Trigonometric relationships in triangles are explored, showing how sine and cosine functions relate sides and angles:

Example: In a right triangle, a = c * sin α = c * cos β, where c is the hypotenuse and α and β are the non-right angles.

The page also covers important lines within triangles:

Vocabulary: A median in a triangle is a line segment that connects a vertex to the midpoint of the opposite side.

Highlight: The medians of a triangle intersect at a single point called the centroid, which divides each median in a 2:1 ratio.

Lastly, the page touches on the law of sines and provides formulas for the circumradius of a triangle.