Special Lines in a Triangle

Side

Side of a triangle is a line segment that connects two vertices. Triangle has three sides.

Vertex

Vertex is the point of intersection of two sides of triangle.

Median, m

A median of a triangle is a segment from vertex to the midpoint of the opposite side. A triangle has three medians, and these three will intersect at the centroid.

Length of Median: (Apollonius)

Properties

Some fascinating properties of the medians of a triangle:

1. The fact that the three medians always meet at a single point is interesting in its own right.
2. Each median divides the triangle into two smaller triangles which have the same area.
3. The centroid (point where they meet) is the center of gravity of the triangle.
4. The three medians divide the triangle into 6 smaller triangles that all have the same area, even though they may have different shapes.

Angle Bisector

An Angle Bisector of a triangle is a segment or ray that bisects an angle and extends to the opposite side. It is drawn from vertex to the opposite side of the triangle. Since there are three included angles of the triangle, there are also three angle bisectors, and these three will intersect at the incenter.

Length of Angle Bisector:

Altitude, h

An altitude of a triangle is a segment from a vertex perpendicular to the opposite side. The altitudes of the triangle will intersect at a common point called orthocenter.

Length of Altitude:

Midsegment

Midsegment of a Triangle is a line segment joining the midpoints of two sides of a triangle. A triangle has 3 possible midsegments.

Properties

1. The midsegment is always parallel to the third side of the triangle. In the figure above, drag any point around and convince yourself that this is always true.
2. The midsegment is always half the length of the third side. In the figure above, drag point A around. Notice the midsegment length never changes because the side BC never changes.
3. A triangle has three possible midsegments, depending on which pair of sides is initially joined.

Perpendicular Bisector

Perpendicular bisector of the triangle is a perpendicular line that crosses through midpoint of the side of the triangle. The three perpendicular bisectors are worth noting for it intersects at the center of the circumscribing circle of the triangle. The point of intersection is called the circumcenter.

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