Parts Of A Triangle

Aside from its vertices, sides (legs and base), interior and exterior angles, a triangle has also the following parts:

• median
• altitude
• perpendicular bisector of its sides
• angle bisector

These parts are discussed in the following problem.

The Problem

Given isosceles triangle ABC with vertices at A(-3, 4), B(5, 0) and C(1, -4), show analytically that the perpendicular bisector of its base passes through its vertex angle.

Discussion

The illustration above shows isosceles triangle ABC with point X as the midpoint of side BC which is the base.

Given the definitions of the following parts:

• The median is the line formed by a triangle's vertex and the midpoint of the side opposite to that vertex.
• The altitude is the perpendicular line drawn from a triangle's vertex to the side opposite to that vertex (or to the extension of the side opposite to that vertex).
• The perpendicular bisector of a triangle's side divides that side into two equal line segments.
• The angle bisector is a line that divides an angle into two equal angles.

line segment AX is all of the above.

The solution to the above problem serves as a coordinate proof of the theorem: the perpendicular bisector of the base of an isosceles triangle divides the vertex angle into two equal angles.

Solution

In order to show that the perpendicular bisector of the base passes through the vertex angle, it must be shown that line segment AX:

• is perpendicular to the base AB;
• divides base AB into two equal line segments;
• divides angle A into two equal angles.

Distance Between Two Points

• distance_AB = distance_AC = √ 80
• distance_BC = √ 32
• distance_BX = distance_CX = √ 8

Equations of the Line Segments

• line segment AB: x + 2y - 5 = 0; x-intercept = 5; y-intercept = 5/2
• line segment BC: x - y - 5 = 0; x-intercept = 5; y-intercept = - 5
• line segment AC: 2x + y + 2 = 0; x-intercept = - 1; y-intercept = - 2
• line segment AX: x + y - 1 = 0; x-intercept = 1; y-intercept = 1

Slopes of the Line Segments

• m_AB = - 1/2
• m_BC = 1
• m_AC = - 2
• m_AX = - 1

Interior Angles of the Triangle

• vertex angle A = 36.87°
• base angle B = base angle C = 71.56°
• angle CAX = angle BAX = 18.43°

Medians And Altitudes of Isosceles Triangles

Given the same isosceles triangle ABC:

• point D is the midpoint of leg AB;
• point E is the midpoint of leg AC;
• point F is the intersection of leg AB and altitude CF;
• point G is the intersection of leg AC and altitude BG.

Line segments CD and BE are medians to the legs of triangle ABC.



The graph above shows that for any isosceles triangle, its medians to the legs are congruent.

Distance Between Two Points

• distance_AD = distance_BE = distance_CD = distance_CE = √ 20
• distance_BE = distance_CD = √ 36
• distance_BG = distance_CF = √ 720/25

Equations of the Medians And the Altitudes

• median CD: x = 1
• median BE: y = 0
• altitude CF: 2x - y - 6 = 0
• altitude BG: x - 2y - 5 = 0

Coordinates of:

• point F: (17/5, 4/5)
• point G: (1/5, - 12/5)

This last graph illustrates that for any isosceles triangle, its altitudes to the legs are congruent.