Incentre of a triangle formula and properties

Learn what is the meaning of incentre of a triangle and also learn its important properties. As it is an important concept in coordinate geometry so you have to learn it.

Incentre of a triangle

Incentre of a triangle definition is given as “the point of intersection of internal angular bisectors of a triangle”. Here we used the word angular bisector. Do you know what does it mean? It means that the line which divides the angle into two equal angles.

The Formula for finding incentre of a triangle:

Incentre of a triangle

Let PQR be a triangle with sides PQ = r, QR = p, RP = q.

Therefore, Incentre of triangle PQR is given as

Incentre of a triangle


 It should be noted that the side which is opposite to vertex P should be represented by the letter p and the side opposite to vertex Q should be represented by the letter q and finally, the side opposite to vertex R should be represented by the letter r. This notation you have to follow for any triangle. The triangle may be ABC or PQR or DEF or any other triangle we should follow the above pattern.

Do you know what is r? Yes, it is the distance from P to Q. similarly p is the distance from point Q to R and q is the distance from point R to P. All these p, q, r lengths we can find them by using the distance between two points formula.

The important Properties of incentre of a triangle:

→For any given triangle the point of intersection of all the angular bisectors is called as incentre”.

→It is clear that the incentre of any triangle will always lie inside the triangle.

→Incentre and the centre of incircle coincide each other that means both are same.

→Incentre of a triangle = centre of the incircle of the triangle.

→The perpendicular distance from the three sides of a triangle to the incentre will be always equal.

Incentre of a triangle

→From the above triangle with incentre I, we can say that

    EI = FI= DI

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