The Triangle and its Properties - Medians and Altitudes of a Triangle

A median of a triangle is a line segment that joins a vertex of the triangle to the midpoint of the opposite side. Visually, if you have a triangle $\triangle ABC$ and you find the exact middle point $D$ on the side $BC$, the line $AD$ is the median. It essentially bisects the side into two equal parts such that $BD = DC$.

Every triangle has exactly three medians, one from each vertex. These three medians always intersect at a single point inside the triangle called the Centroid. Visually, the centroid can be thought of as the 'balancing point' or center of gravity of the triangle.

An altitude of a triangle is a perpendicular line segment drawn from a vertex to the line containing the opposite side. Visually, an altitude represents the 'height' of the triangle. It always meets the opposite side (or its extension) at a right angle, which is $90^{\circ}$.

A triangle has three altitudes, one for each vertex. The point where all three altitudes meet is called the Orthocenter. Unlike the centroid, the orthocenter's location changes visually based on the triangle type: it stays inside for acute triangles, sits at the right-angle vertex for right triangles, and falls outside for obtuse triangles.

In an obtuse-angled triangle, two of the altitudes lie outside the triangle. To visualize this, you must extend the base lines of the triangle outward so that the perpendicular line from the opposite vertex can meet them.

In an isosceles triangle, the median and altitude drawn from the vertex connecting the two equal sides are the same line segment. In an equilateral triangle, all three medians are identical to the three altitudes, meaning the Centroid and the Orthocenter are the exact same point at the very center of the triangle.