What is the difference between medians, perpendiculars bisectors, and altitudes?

Medians, perpendicular bisectors, and altitudes are geometric constructs used in triangles. Each serves a different purpose and has unique properties. Here's a detailed explanation of each:

Median:

Definition: A median of a triangle is a line segment that joins a vertex to the midpoint of the opposite side.
Properties:
It divides the triangle into two smaller triangles of equal area.
The three medians of a triangle intersect at a point called the centroid.
The centroid is the center of mass or balance point of the triangle.
The centroid divides each median into two parts in the ratio 2:1, with the longer segment being closer to the vertex.
Example: In a triangle ABC, if D is the midpoint of side BC, the line segment AD is a median.
Perpendicular Bisector:

Definition: A perpendicular bisector of a side of a triangle is a line that is perpendicular to the side and passes through its midpoint.
Properties:
It is not required to pass through a vertex of the triangle.
The three perpendicular bisectors of a triangle meet at a point called the circumcenter.
The circumcenter is equidistant from all three vertices of the triangle and is the center of the circumcircle, the circle that passes through all three vertices of the triangle.
The circumcenter may lie inside, outside, or on the triangle, depending on whether the triangle is acute, obtuse, or right-angled.
Example: In a triangle ABC, the perpendicular bisector of side BC is a line that is perpendicular to BC and passes through its midpoint.
Altitude:

Definition: An altitude of a triangle is a perpendicular line segment drawn from a vertex to the opposite side (or its extension).
Properties:
It represents the shortest distance from the vertex to the opposite side.
The three altitudes of a triangle intersect at a point called the orthocenter.
The orthocenter may lie inside, outside, or on the triangle, depending on whether the triangle is acute, obtuse, or right-angled.
Example: In a triangle ABC, if a perpendicular is drawn from vertex A to side BC (or its extension), that line is an altitude.
Key Differences:

A median connects a vertex to the midpoint of the opposite side, dividing the triangle into two equal-area triangles.
A perpendicular bisector is perpendicular to a side and passes through its midpoint but does not necessarily pass through a vertex.
An altitude is a perpendicular segment from a vertex to the opposite side (or its extension), representing the height of the triangle.
Each construct has a unique point of concurrency (centroid, circumcenter, orthocenter) and distinct applications in geometry.