Circumcircle, Incircle & Triangle Properties: JEE Main

The properties of circles associated with triangles — circumcircle, incircle, and excircles — form a rich set of interrelated formulas that JEE Main tests through direct calculation and through more subtle relationship questions. This guide gives you all the key formulas, their derivations in brief, and the question types associated with each.

Circumcircle: Radius and Center

The circumcircle passes through all three vertices of a triangle. Circumradius: R = a/(2 sinA) = b/(2 sinB) = c/(2 sinC) (sine rule). Also: R = abc/(4K) where K = area of triangle. For area K = (1/2)ab sinC: R = abc/(4 × ½ab sinC) = c/(2 sinC) ✓. For a right triangle (C = 90°): R = c/2 (hypotenuse is the diameter). Circumcentre location: interior of acute triangle; on the hypotenuse of right triangle; exterior of obtuse triangle. JEE tests: find R given the sides, or find a side given R and an angle. The formula R = a/(2 sinA) connects R to both side length and opposite angle — a two-step bridge between geometry and trigonometry.

Euler's formula: distance between circumcentre O and incentre I: OI² = R(R − 2r) where r is the inradius. Corollary: R ≥ 2r (Euler's inequality), with equality for equilateral triangle. JEE has asked for OI² given R and r directly. Take a free trigonometry and geometry mock. See our properties of triangles guide.

Incircle and Inradius

The incircle is tangent to all three sides of a triangle. Inradius: r = K/s where K = area and s = (a+b+c)/2 = semi-perimeter. Also: r = (s−a) tanA/2 = (s−b) tanB/2 = (s−c) tanC/2. And: r = 4R sinA/2 sinB/2 sinC/2. The most useful forms: r = K/s and r = 4R sin(A/2)sin(B/2)sin(C/2). Area K = rs (inradius × semi-perimeter — a clean formula for area when the incircle is given). JEE question: triangle with perimeter 12 and area 6 → s = 6, r = 6/6 = 1. Or: given R = 5, r = 2, find OI² = R(R−2r) = 5(5−4) = 5 → OI = √5.

Incentre coordinates: I = (aA + bB + cC)/(a + b + c) where A, B, C are the vertex coordinates (a, b, c are opposite side lengths). This is the weighted average of vertices with weights = opposite side lengths.

Excircles and Exradii

The excircle opposite to vertex A (radius r_A) is tangent to side a and to extensions of sides b and c. Exradius: r_A = K/(s−a), r_B = K/(s−b), r_C = K/(s−c). Also: r_A = s tan(A/2), r_B = s tan(B/2), r_C = s tan(C/2). Relations: r × r_A × r_B × r_C = K² and 1/r = 1/r_A + 1/r_B + 1/r_C. JEE tests the exradius formulas in substitution questions: given a, b, c (or equivalently K and s), find r_A; or given r, r_A, r_B, find r_C from the harmonic relation. The formula r × r_A × r_B × r_C = K² connects all four circles to the area — elegant and directly testable.

Special Triangles and Key Identities

Equilateral triangle (a = b = c = a, A = B = C = 60°): R = a/√3, r = a/(2√3), OI = 0 (circumcentre = incentre). Regular tetrahedron analog: R = 2r (only for equilateral, since Euler's inequality R ≥ 2r becomes equality). Right triangle (C = 90°): R = c/2 (largest side is diameter), r = (a + b − c)/2 (derived from r = K/s with K = ab/2, s = (a+b+c)/2). The right-triangle inradius formula r = (a + b − c)/2 is a standard JEE substitution question. Pedal triangle: the triangle formed by the feet of the altitudes of an acute triangle has circumradius R/2, where R is the circumradius of the original triangle — tested as a JEE conceptual question. For circle theorems see our circles guide and for area of triangles using trigonometry see our trigonometry guide.