Bisector, median, height, and midline of a triangle

Bisector of a triangle

The bisector of a triangle is a segment of the bisector of an angle of a triangle that connects the vertex of the triangle to a point on the opposite side.

1. The bisectors of any triangle intersect at one point.
2. The bisector of an angle of a triangle divides the opposite side into parts proportional to the adjacent sides.

AB ∕ AC = A₁B ∕ A₁C.

3. In a triangle, the intersection point of the bisectors divides the bisector in the ratio

AO ∕ OA₁=(AB+AC) ∕ BC.

4. The length of the bisector l_с, drawn to the side c of a triangle with sides a, b, c is equal to where c_a and c_b – segments into which the bisector divides the side c.

Median of a triangle

The median of a triangle is a segment that connects the vertex of the triangle with the midpoint of the opposite side.

5. The medians of any triangle intersect at one point.
6. In a triangle, the intersection point of the medians divides the medians in a ratio of 2:1, counting from the vertex.

7. The length of the median m_a, drawn to the side a of a triangle with sides a, b, c is equal to
8. The median divides a triangle into two triangles of equal area.
9. Three medians divide a triangle into 6 triangles of equal area.

Height of a triangle

The height of a triangle is a perpendicular that is lowered from the vertex of the triangle onto a straight line containing the opposite side of the triangle. The base of this perpendicular is also called the height base.

10. The heights of the triangle (or their extensions) intersect at a one point. This point is called the orthocenter of the triangle. In an obtuse triangle, the orthocenter is outside the triangle.

An orthocentric triangle

An orthocentric triangle is a triangle whose vertices are the bases of the heights of the original triangle.

11. The heights of an acute triangle are the bisectors of the angles of its orthocentric triangle.

The middle line of a triangle

The midline of a triangle is the segment that connects the midpoints of its two sides.

12. The midline of a triangle, connecting the midpoints of two of its sides, is parallel to the third side and equal to half of it.

13. The three middle lines of a triangle divide the triangle into 4 equal triangles.