The Exterior Angle Bisectors and the Excenters of a Triangle

The exterior angle bisectors of a triangle intersect at three points called excenters.

These points of intersection are the centers of the circles tangent to the sides and the side extensions of the triangle.

This post is the last part of a series of posts about the points of concurrency of a triangle's parts.

The Problem

Given an isosceles triangle ABC with vertices at A(-2, 2), B(2, 0) and C(0, -2), find the points of intersection of its three exterior angle bisectors. Find the equations of the circles tangent to the sides of the triangle.

Discussion

In the given illustration above, line segments DE, EF and DF are the bisectors of the exterior angles of isosceles triangle ABC.

Their points of intersection (point D, point E and point F) are called excenters because it is the centers of the circles tangent to the sides and side extensions of triangle ABC.

Points M to U are the points of tangency.

Solution

Centers of the Circles

• circle D (1756/1511, 7800/1511)
• circle E (18132000/7594697, -18129800/7594697)
• circle F (- 36000/6973, - 8108/6973)

Radii of the Circles

• circle D : radius DM = radius DN = radius DO = √ 18
• circle E : radius EP = radius EQ = radius ER = √ 38/10
• circle F : radius FS = radius FT = radius FU = √ 18

Points of Tangency

• point M (- 92/35, 114/35)
• point N (- 5554/7555, 10332/7555)
• point O (6289/1511, 3267/1511)
• point P (123976994/37973485, - 24015012/37973485)
• point Q (690527/690427, - 690327/690427)
• point R (24012812/37973485, -123972594/37973485)
• point S (-15081/6973, - 29027/6973)
• point T (- 47676/34865, 25622/34865)
• point U (- 113838/34865, 91784/34865)

Distance Between Two Points

• side_AB = side_AC = √ 20
• side_BC = √ 8
• line segment_AM = distance_AN = √ 2
• distance_AT = distance_AU = √ 2
• distance_BN = distance_BO = √ 9
• distance_BP = distance_BQ = √ 2
• distance_CQ = distance_CR = √ 2
• distance_CS = distance_CT = √ 9

Equations of the Line Segments

• line AB: x + 2y - 2 = 0
• line BC: x - y - 2 = 0
• line AC: 2x + y + 2 = 0
• line DF: x - y + 4 = 0
• line DE: 1300x + 211y - 2600 = 0
• line EF: 973x + 6000y + 12000 = 0

Equation of the Circles

• Circle D: ( x - 1756/1511 )² + ( y - 7800/1511 )² = 18
• Circle E: ( x - 18132000/7594697 )² + ( y + 18129800/7594697 )² = 38/10
• Circle F: ( x + 36000/6973 )² + ( y + 8108/6973 )² = 18

Slopes of the Line Segments

• m_AB = - 1/2
• m_BC = 1
• m_AC = - 2
• m_DE = - 1300/211
• m_DF = 1
• m_EF = - 973/6000

Interior Angles of the Triangle

• angle NAT = 36.87°
• angle NBQ = 71.56°
• angle QCT = 71.56°

Exterior Angles of the Triangle

• angle MAN = angle TAU = 143.13°
• angle NBO = angle PBQ = 108.44°
• angle QCR = angle SCT = 108.44°
• ½ angle MAN = ½ angle TAU = angle DAM = angle DAN = angle FAT = angle FAU
• ½ angle NBO = ½ angle PBQ = angle DBN = angle DBO = angle EBP = angle EBQ
• ½ angle QCR = ½ angle SCT = angle ECQ = angle ECR = angle FCS = angle FCT