Triangle Excenter Calculator
Calculate triangle excenter instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Triangle Excenter Calculator
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Formula: r_A = Area / (s - a) | r_B = Area / (s - b) | r_C = Area / (s - c)
Worked example — r_A = 3.6742 | r_B = 4.8990 | r_C = 7.3485 | Inradius = 1.6330
Formula
r_A = Area / (s - a) | r_B = Area / (s - b) | r_C = Area / (s - c)
Where r_A, r_B, r_C are the exradii opposite vertices A, B, C respectively. Area is calculated using Heron formula, and s is the semi-perimeter (a+b+c)/2. Each exradius is found by dividing the area by the difference of the semi-perimeter and the opposite side.
Worked Examples
Example 1: Finding All Three Exradii
Problem:A triangle has sides a = 5, b = 6, c = 7. Calculate all three exradii and the inradius.
Solution:Semi-perimeter s = (5 + 6 + 7) / 2 = 9 Area = sqrt(9 x 4 x 3 x 2) = sqrt(216) = 14.6969 Inradius r = Area / s = 14.6969 / 9 = 1.6330 Exradius r_A = Area / (s - a) = 14.6969 / 4 = 3.6742 Exradius r_B = Area / (s - b) = 14.6969 / 3 = 4.8990 Exradius r_C = Area / (s - c) = 14.6969 / 2 = 7.3485 Verification: 1/r = 1/3.6742 + 1/4.8990 + 1/7.3485 = 0.6124 = 1/1.6330
Result:r_A = 3.6742 | r_B = 4.8990 | r_C = 7.3485 | Inradius = 1.6330
Example 2: Excenter Coordinates for a Right Triangle
Problem:A right triangle has vertices at A(0,0), B(4,0), C(0,3). Find the excenter coordinates.
Solution:Sides: a = BC = 5, b = AC = 3, c = AB = 4 s = (5 + 3 + 4) / 2 = 6, Area = (4 x 3)/2 = 6 Excenter opposite A: I_A = (-5(0,0) + 3(4,0) + 4(0,3)) / (-5+3+4) I_A = (12, 12) / 2 = (6, 6) Excenter opposite B: I_B = (5(0,0) - 3(4,0) + 4(0,3)) / (5-3+4) I_B = (-12, 12) / 6 = (-2, 2) Excenter opposite C: I_C = (5(0,0) + 3(4,0) - 4(0,3)) / (5+3-4) I_C = (12, -12) / 4 = (3, -3)
Result:I_A = (6, 6) | I_B = (-2, 2) | I_C = (3, -3)
Frequently Asked Questions
What is an excenter of a triangle?
An excenter of a triangle is the center of an excircle, which is a circle that is tangent to one side of the triangle and to the extensions of the other two sides. Every triangle has exactly three excenters, one opposite each vertex. The excenter opposite vertex A (denoted I_A) is the point where the external bisector of angle B, the external bisector of angle C, and the internal bisector of angle A all meet. Each excircle lies entirely outside the triangle itself. The three excenters together with the incenter form a special quadrilateral called the excentral triangle. Excenters are important in advanced geometry, particularly in the study of triangle centers and the relationships between different circles associated with a triangle.
How do you calculate the exradius of a triangle?
The exradius is the radius of an excircle, and there is a simple formula for each of the three exradii. The exradius opposite vertex A is r_A = Area / (s - a), where s is the semi-perimeter and a is the side opposite vertex A. Similarly, r_B = Area / (s - b) and r_C = Area / (s - c). The area can be found using Heron formula: Area = sqrt(s(s-a)(s-b)(s-c)). For example, with sides 5, 6, 7: s = 9, Area = sqrt(9 x 4 x 3 x 2) = sqrt(216) = 14.6969. Then r_A = 14.6969 / (9-5) = 3.6742, r_B = 14.6969 / (9-6) = 4.8990, and r_C = 14.6969 / (9-7) = 7.3485. The exradius opposite the longest side is always the largest.
What is the relationship between the inradius and exradii?
There are several elegant relationships between the inradius r and the three exradii r_A, r_B, r_C. One fundamental identity is: 1/r = 1/r_A + 1/r_B + 1/r_C. Another important relationship is: r_A + r_B + r_C = r + 4R, where R is the circumradius. The area of the triangle can be expressed as Area = r times s = r_A times (s - a) = r_B times (s - b) = r_C times (s - c). Also, r times r_A times r_B times r_C = Area squared. These relationships demonstrate the deep interconnections between the incircle, excircles, and circumcircle of a triangle, and they are frequently used in competition mathematics and advanced geometric proofs to derive other properties.
How do you find the coordinates of an excenter?
The coordinates of the excenters can be found using weighted formulas based on the side lengths and vertex positions. If the vertices have coordinates A(x1,y1), B(x2,y2), C(x3,y3) and the opposite sides have lengths a, b, c respectively, then the excenter opposite A is I_A = (-a*A + b*B + c*C) / (-a + b + c). Similarly, I_B = (a*A - b*B + c*C) / (a - b + c), and I_C = (a*A + b*B - c*C) / (a + b - c). Notice the pattern: one sign is negative, corresponding to the vertex being excluded. The incenter formula uses all positive signs: I = (a*A + b*B + c*C) / (a + b + c). These formulas show that excenters are essentially signed weighted averages of the vertex positions.
What is the excentral triangle?
The excentral triangle is formed by connecting the three excenters I_A, I_B, and I_C of a triangle. It has several remarkable properties. The original triangle is the medial triangle of the excentral triangle, meaning the original vertices are the midpoints of the excentral triangle sides. The incenter of the original triangle is the orthocenter of the excentral triangle. The circumradius of the excentral triangle is 2R (twice the circumradius of the original triangle). The sides of the excentral triangle are perpendicular to the angle bisectors of the original triangle. The area of the excentral triangle is always larger than the original triangle by a factor related to the cosines of the half-angles.
What is an excircle and how does it differ from the incircle?
An excircle is a circle that is tangent to one side of the triangle from outside and tangent to the extensions of the other two sides. The incircle, by contrast, is tangent to all three sides from the inside and lies entirely within the triangle. Each triangle has exactly one incircle but three excircles. The incircle has its center (incenter) at the intersection of the three internal angle bisectors, while each excircle has its center (excenter) at the intersection of one internal and two external angle bisectors. The inradius is always smaller than any of the three exradii. The incircle and excircles together are called the four tritangent circles of the triangle, and they satisfy beautiful symmetric relationships involving the triangle area and semi-perimeter.
What is the Nagel point and how does it relate to excenters?
The Nagel point is a triangle center closely related to the excircles. It is defined as the point where the three lines connecting each vertex to the point of tangency of the opposite excircle with the opposite side all intersect. If the excircle opposite A touches side BC at point T_A, the excircle opposite B touches side AC at T_B, and the excircle opposite C touches side AB at T_C, then lines A-T_A, B-T_B, and C-T_C are concurrent at the Nagel point. The Nagel point has the property that the distances from it to each side are related to the exradii. It is the isotomic conjugate of the Gergonne point (which is defined similarly using the incircle tangent points) and lies on the line connecting the incenter to the centroid.
How are excircles used in solving geometry competition problems?
Excircles appear frequently in mathematical olympiad problems and geometry competitions because they create elegant relationships. Common techniques include using the tangent length properties: the tangent from vertex A to the excircle opposite A has length s (the semi-perimeter), while tangent from B or C to the same excircle has length s minus the adjacent side. These tangent lengths help set up equations for finding unknown sides. The Euler formula OI squared = R squared minus 2Rr (relating circumcenter O, incenter I, circumradius R, and inradius r) has excircle analogues: OI_A squared = R squared + 2R times r_A. Competition problems often involve proving that certain points are concyclic using excircle properties or that certain lines are concurrent through excenter constructions.
What is the Feuerbach theorem and its connection to excircles?
The Feuerbach theorem states that the nine-point circle of any triangle is internally tangent to the incircle and externally tangent to all three excircles. This is one of the most beautiful results in classical geometry. The nine-point circle passes through the midpoints of the three sides, the feet of the three altitudes, and the midpoints of the segments from the vertices to the orthocenter. The point of tangency between the nine-point circle and the incircle is called the Feuerbach point. This theorem reveals a deep connection between seemingly unrelated circles associated with a triangle. The proof typically uses inversive geometry or analytic methods and demonstrates why the nine-point circle has radius exactly R/2 where R is the circumradius.
Can excircles be constructed using compass and straightedge?
Yes, excircles can be constructed using classical compass and straightedge methods. To construct the excircle opposite vertex A, first extend sides AB and AC beyond B and C respectively. Then bisect the exterior angles at B and C using the standard angle bisection construction. The intersection of these two external angle bisectors gives the excenter I_A. From I_A, drop a perpendicular to side BC (or to either extended side) to find the exradius. Then draw a circle centered at I_A with this radius. The construction is identical in principle to finding the incenter using internal bisectors, except you use external bisectors at two vertices. This construction was known to ancient Greek geometers and remains a standard exercise in classical geometry courses and textbooks.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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