Excenter of a Triangle (original) (raw)

Last Updated : 23 Jul, 2025

What is the Excenter of Triangle?

The excenter of a triangle is the center of an excircle, which is a circle tangent to one side of the triangle and the extensions of the other two sides. Each triangle has three excenters, corresponding to each vertex. They lie at the intersection of the angle bisectors of the internal and external angles opposite the respective vertices.

Formula for Excenter of a Triangle

The excenters of a triangle can be found using the following formulas based on the triangle's vertices A(x1, y1), B(x2, y2), and C(x3, y3).

**For excenter I_A opposite vertex A: I_A = ((-a x1 + b**x2 + c x3) / (-a + b + c), (-a**y1 + b y2 + c**y3) / (-a + b + c))

**For excenter I_B opposite vertex B: I_B = ((a x1 - b**x2 + c x3) / (a - b + c), (a**y1 - b y2 + c**y3) / (a - b + c))

**For excenter I_C opposite vertex C: I_C = ((a x1 + b**x2 - c x3) / (a + b - c), (a**y1 + b y2 - c**y3) / (a + b - c))

where a, b, and c are the lengths of the sides opposite the vertices A, B, and C respectively.

Properties of Excenter of a Triangle

• Each excenter is the center of an excircle tangent to one side of the triangle and the extensions of the other two sides.
• The excenter is the intersection point of the internal angle bisector of one vertex and the external angle bisectors of the other two vertices.
• The three excenters and the orthocenter of the triangle are collinear, lying on the Euler line.
• The incenter and the three excenters form an orthocentric system, with the incenter as the orthocenter of the triangle formed by the excenters.
• The excenters are the vertices of the cevian triangle, formed by drawing cevians from each vertex to the opposite side that are concurrent.

**Related Articles:

• Types of Center in a Triangle
• Circumcenter of Triangle

Excenter of a Triangle- Solved Problems

**Question 1: In triangle ABC, I_A is the excenter opposite vertex A. If AB = 5, AC = 7, and BC = 6, find the length of AI_A.

**Solution 1: To find AI_A, we can use the formula for the length from a vertex to the corresponding excenter: AI_A = r / cos(A/2) where r is the radius of the incircle and A/2 is half of angle A.

First, find the semi-perimeter s: s = (AB + AC + BC) / 2 = (5 + 7 + 6) / 2 = 9

Next, calculate the area Δ of triangle ABC using Heron's formula: Δ = √(s * (s - AB) * (s - AC) * (s - BC)) = √(9 * (9 - 5) * (9 - 7) * (9 - 6)) = √(9 * 4 * 2 * 3) = √216 = 6√6

Now, find the radius r of the incircle: r = Δ / s = (6√6) / 9 = (2√6) / 3

Calculate cos(A/2): cos(A/2) = √(s * (s - AB) / (AB * AC)) = √(9 * 4 / (5 * 7)) = √(36 / 35)

Finally, compute AI_A: AI_A = r / cos(A/2) = (2√6 / 3) / √(36 / 35) = (2√6 / 3) * (√35 / 6) = √210 / 9

Therefore, the length of AI_A is √210 / 9.

**Question 2: In triangle XYZ, I_B is the excenter opposite vertex B. If XY = 8, XZ = 10, and YZ = 6, find the length of BI_B.

**Solution 2: Using a similar approach as in Solution 1:

Calculate the semi-perimeter s: s = (XY + XZ + YZ) / 2 = (8 + 10 + 6) / 2 = 12

Find the area Δ of triangle XYZ: Δ = √(s * (s - XY) * (s - XZ) * (s - YZ)) = √(12 * (12 - 8) * (12 - 10) * (12 - 6)) = √(12 * 4 * 2 * 6) = 4√36 = 24

Calculate the radius r of the incircle: r = Δ / s = 24 / 12 = 2

Find cos(B/2): cos(B/2) = √(s * (s - XY) / (XY * XZ)) = √(12 * 6 / (8 * 10)) = √(72 / 80) = √0.9

Compute BI_B: BI_B = r / cos(B/2) = 2 / √0.9 = 2√10 / 3

Therefore, the length of BI_B is 2√10 / 3.

**Question 3: In triangle PQR, I_C is the excenter opposite vertex R. If PQ = 12, PR = 15, and QR = 9, find the length of RI_C.

**Solution 3: To find RI_C, we use the formula: RI_C = r / cos(C/2), where r is the radius of the incircle and C/2 is half of angle C.

First, calculate the semi-perimeter s: s = (PQ + PR + QR) / 2 = (12 + 15 + 9) / 2 = 18

Next, calculate the area Δ of triangle PQR using Heron's formula: Δ = √(s * (s - PQ) * (s - PR) * (s - QR)) = √(18 * (18 - 12) * (18 - 15) * (18 - 9)) = √(18 * 6 * 3 * 9) = √(2916) = 54

Now, find the radius r of the incircle: r = Δ / s = 54 / 18 = 3

Calculate cos(C/2): cos(C/2) = √(s * (s - PQ) / (PQ * PR)) = √(18 * 6 / (12 * 15)) = √(108 / 180) = √(3 / 5)

Finally, compute RI_C: RI_C = r / cos(C/2) = 3 / √(3 / 5) = 3√(5 / 3) = √15

Therefore, the length of RI_C is √15.

**Question 4: In triangle LMN, I_A is the excenter opposite vertex L. If LM = 10, LN = 12, and MN = 14, find the length of LI_A.

**Solution 4: To find LI_A, we use the formula: LI_A = r / cos(A/2), where r is the radius of the incircle and A/2 is half of angle A.

First, calculate the semi-perimeter s: s = (LM + LN + MN) / 2 = (10 + 12 + 14) / 2 = 18

Next, calculate the area Δ of triangle LMN using Heron's formula: Δ = √(s * (s - LM) * (s - LN) * (s - MN)) = √(18 * (18 - 10) * (18 - 12) * (18 - 14)) = √(18 * 8 * 6 * 4) = √(3456) = 24√6

Now, find the radius r of the incircle: r = Δ / s = (24√6) / 18 = (4√6) / 3

Calculate cos(A/2): cos(A/2) = √(s * (s - LM) / (LM * LN)) = √(18 * 8 / (10 * 12)) = √(144 / 120) = √(6 / 5)

Finally, compute LI_A: LI_A = r / cos(A/2) = (4√6 / 3) / √(6 / 5) = (4√6 / 3) * (√(5 / 6)) = 4√(5 / 2)

Therefore, the length of LI_A is 4√(5 / 2).

**Question 5: In triangle XYZ, I_C is the excenter opposite vertex Z. If XZ = 8, YZ = 15, and XY = 17, find the length of ZI_C.

**Solution 5: To find ZI_C, we use the formula: ZI_C = r / cos(C/2), where r is the radius of the incircle and C/2 is half of angle C.

First, calculate the semi-perimeter s: s = (XY + YZ + XZ) / 2 = (17 + 15 + 8) / 2 = 20

Next, calculate the area Δ of triangle XYZ using Heron's formula: Δ = √(s * (s - XY) * (s - YZ) * (s - XZ)) = √(20 * (20 - 17) * (20 - 15) * (20 - 8)) = √(20 * 3 * 5 * 12) = √(3600) = 60

Now, find the radius r of the incircle: r = Δ / s = 60 / 20 = 3

Calculate cos(C/2): cos(C/2) = √(s * (s - XY) / (XY * YZ)) = √(20 * 3 / (17 * 15)) = √(60 / 255) = √(12 / 51)

Finally, compute ZI_C: ZI_C = r / cos(C/2) = 3 / √(12 / 51) = 3√(51 / 4)

Therefore, the length of ZI_C is 3√(51 / 4).

Practice Problems on Excenter of a Triangle

1. In triangle ABC, I_A is the excenter opposite vertex A. If AB = 8, AC = 10, and BC = 6, find the length of AI_A.
2. In triangle PQR, I_B is the excenter opposite vertex Q. If PQ = 12, PR = 15, and QR = 9, find the length of QI_B.
3. In triangle XYZ, I_C is the excenter opposite vertex Z. If XY = 5, XZ = 13, and YZ = 12, find the length of ZI_C.
4. In triangle LMN, I_A is the excenter opposite vertex L. If LM = 9, LN = 7, and MN = 8, find the length of LI_A.
5. In triangle DEF, I_B is the excenter opposite vertex E. If DE = 11, DF = 14, and EF = 9, find the length of EI_B.
6. In triangle UVW, I_C is the excenter opposite vertex U. If UV = 6, UW = 10, and VW = 8, find the length of UI_C.
7. In triangle STU, I_A is the excenter opposite vertex S. If ST = 15, SU = 12, and TU = 9, find the length of SI_A.
8. In triangle XYZ, I_B is the excenter opposite vertex Y. If XY = 7, XZ = 10, and YZ = 12, find the length of YI_B.
9. In triangle ABC, I_C is the excenter opposite vertex C. If AB = 13, AC = 12, and BC = 5, find the length of CI_C.
10. In triangle DEF, I_A is the excenter opposite vertex D. If DE = 6, DF = 8, and EF = 10, find the length of DI_A.

Conclusion - Excenter of a Triangle

The excenters of a triangle, found at the intersections of internal and external angle bisectors, play crucial roles in geometry, contributing to properties like collinearity and forming orthocentric systems.