the median, altitude and angle bisector of a triangle

From a vertex of a triangle you can draw several special line segments to (or toward) the opposite side, and three of them have names worth knowing. The median goes from a vertex to the midpoint of the opposite side. The altitude goes from a vertex perpendicular to the opposite side (or its extension) — it is the triangle's height from that vertex. The angle bisector goes from a vertex along the line that splits the vertex angle into two equal halves.

Precisely, from vertex A of triangle ABC: the median is the segment from A to the midpoint of BC; the altitude is the segment from A meeting line BC at a right angle; the angle bisector is the segment from A that divides angle A into two equal angles. In general these are three different segments. But in an isosceles triangle, taken from the apex (the vertex between the two equal sides), all three coincide into a single line — which is also the perpendicular bisector of the base — a tidy fact that powers many proofs. A segment from a vertex to the opposite side is called a cevian, so all three are cevians of special type.

Do not assume the three always coincide; that happens only at the apex of an isosceles triangle (and at every vertex of an equilateral one). In a scalene triangle the median, altitude, and bisector from the same vertex are three distinct segments landing at three different points on the opposite side.