HG--Special Segments in Triangles

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

An altitude is a perpendicular segment from a vertex to the line containing the opposite side.

A midsegment is a segment that connects the midpoints of two sides of a triangle.

An angle bisector is a ray that divides an angle into two equal parts.

A perpendicular bisector is a line that divides a segment into two equal parts at a 90-degree angle.

The circumcenter is the point where the perpendicular bisectors of the sides of a triangle intersect, and it is equidistant from the vertices.

The centroid is the point where the three medians of a triangle intersect, and it is the center of mass.

The length of a median can be found using the formula: m = 1/2 * √(2a² + 2b² - c²), where a and b are the lengths of the sides adjacent to the median, and c is the length of the opposite side.

The circumcenter is equidistant from all three vertices of the triangle.

The centroid divides each median into two segments, with the segment connecting the centroid to the vertex being twice as long as the segment connecting the centroid to the midpoint of the opposite side.