Maths · JEE Advanced only

Incentre of a Triangle (JEE Advanced)

Not in the JEE Main syllabus, but it is in JEE Advanced. Study time: about 2 to 3 hours to learn the concept, formulas, and practice 10 to 15 problems of varying difficulty..

Last updated: 26 October 2026

Direct answer

The incentre of a triangle is the point where all three angle bisectors meet.

The incentre of a triangle is the point where all three angle bisectors meet. It is equidistant from all three sides and serves as the centre of the incircle, the largest circle that fits inside the triangle.

Official syllabus

What the JEE Advanced syllabus says

“Centroid, orthocentre, incentre and circumcentre of a triangle”

Source: JEE Advanced 2026 Information Bulletin, Annexure-I. Closest JEE Main topic: co-ordinate of the centroid, orthocentre and circumcentre of a triangle.

JEE Main covers centroid, orthocentre, and circumcentre but skips incentre. JEE Advanced explicitly includes it, so you need to learn the incentre formula and its properties separately.

Key ideas

What you must understand

The incentre divides the triangle into three smaller triangles of equal inradius but different areas.
The incircle touches each side of the triangle exactly once, and the distance from the incentre to each side equals the inradius r.
The incentre always lies inside the triangle, regardless of whether the triangle is acute, right, or obtuse.
The position of the incentre depends on the side lengths a, b, c (not the angles directly in the coordinate formula).
Area of triangle = r × s, where s is the semi-perimeter and r is the inradius.
The incentre divides the angle bisector in a specific ratio related to the adjacent sides.

Formulas

Formulas and key facts

Incentre I = ((a·x₁ + b·x₂ + c·x₃)/(a+b+c), (a·y₁ + b·y₂ + c·y₃)/(a+b+c)) where a, b, c are side lengths opposite to vertices A(x₁,y₁), B(x₂,y₂), C(x₃,y₃)
Inradius r = Area / s, where s = (a+b+c)/2 is the semi-perimeter
Inradius r = (s−a)tan(A/2) = (s−b)tan(B/2) = (s−c)tan(C/2)
Area of triangle = r × s
Distance from incentre to each side = r (inradius)
Length of tangent from vertex A to incircle = s − a
For a right triangle with legs p, q and hypotenuse h: r = (p + q − h)/2

Avoid

Common mistakes

Confusing the incentre formula with the centroid formula. Centroid uses equal weights (1:1:1), but incentre uses side lengths a, b, c as weights.
Mixing up which side length goes with which vertex. Side a is opposite vertex A, not adjacent to it.
Forgetting that the incentre is always inside the triangle. Unlike the orthocentre or circumcentre, it never lies outside.
Using angle measures instead of side lengths in the coordinate formula for incentre.

Worked example

A JEE Advanced-level example

Question. Find the incentre of the triangle with vertices A(0, 0), B(8, 0), and C(0, 6).

Show solution

Step 1: Find the side lengths. Side a (opposite A, from B to C) = √[(8−0)² + (0−6)²] = √(64+36) = √100 = 10. Side b (opposite B, from A to C) = √[(0−0)² + (0−6)²] = 6. Side c (opposite C, from A to B) = √[(0−8)² + (0−0)²] = 8.
Step 2: Apply the incentre formula. I_x = (a·x₁ + b·x₂ + c·x₃)/(a+b+c) = (10·0 + 6·8 + 8·0)/(10+6+8) = 48/24 = 2.
Step 3: I_y = (a·y₁ + b·y₂ + c·y₃)/(a+b+c) = (10·0 + 6·0 + 8·6)/(24) = 48/24 = 2.
Step 4: Verify using inradius formula. Area = (1/2)×8×6 = 24. Semi-perimeter s = 24/2 = 12. Inradius r = 24/12 = 2. The incentre (2, 2) is indeed at distance 2 from both the x-axis and y-axis (the two legs), confirming our answer.

Answer: Incentre = (2, 2)

Solved twice independently; both solutions agree.

FAQs

Frequently asked questions

Is Incentre of a Triangle in the JEE Main syllabus?

No. It is in the JEE Advanced 2026 syllabus only. The closest JEE Main topic is co-ordinate of the centroid, orthocentre and circumcentre of a triangle.

How long does Incentre of a Triangle take to prepare for JEE Advanced?

About 2 to 3 hours to learn the concept, formulas, and practice 10 to 15 problems of varying difficulty., if you already know the related JEE Main topic.