### Theory:

Incentre: It is the point of concurrency of a triangle formed by the intersection of its three angle bisectors of a triangle. It is denoted by $$I$$.
Incircle:It is the circle drawn inside the triangle with incentre as its centre.
Working rule to construct the Incircle of a Triangle
Case 1:

Given the length of two sides of a triangle and the measure of one of its interior angle.
Example:
Construct the incircle of a triangle $$ABC$$ given $$AB = 8$$ $$cm$$, $$BC = 6$$ $$cm$$ and $$\angle BAC = 55^{\circ}$$. Locate its incentre and find the radius.

Construction:

Step 1: Draw a rough figure for the given question to get the picture of the triangle that is to be constructed.

Step 2: Draw a line segment $$AB = 8$$ $$cm$$ using the ruler.

Step 3: With $$A$$ as centre, mark an angle $$55^{\circ}$$ using the protractor and name it as $$X$$. Now join $$AX$$.

Step 4: With $$B$$ as centre, measure $$6$$ $$cm$$ in the compass and cut an arc intersecting $$AX$$ and mark it as $$C$$.

Step 5: Draw the angle bisectors of any two angles of the triangle $$A$$ and $$B$$ (say).

Step 6: Mark the intersecting point of the angle bisectors as $$I$$, which is the incentre of the given triangle.

Step 7: Draw a perpendicular bisector from $$I$$ to one of the sides of the triangle $$AB$$ (say) to meet $$AB$$ at $$D$$.

Step 8: Measure the length of $$ID$$ to get the inradius of the incircle.

Step 9: With $$I$$ as centre and $$ID$$ as radius, draw the incircle of the triangle. Case 2:

Given the length of a side of a triangle and the measure of two of its interior angle.
Example:
Construct the incircle of a triangle $$ABC$$ given $$AB = 6$$ $$cm$$, $$\angle BAC = 55^{\circ}$$ and $$\angle CBA = 60^{\circ}$$. Locate its incentre and find the radius.

Construction:

Step 1: Draw a rough figure for the given question to get the picture of the triangle that is to be constructed.

Step 2: Draw a line segment $$AB = 6$$ $$cm$$ using the ruler.

Step 3: With $$A$$ as centre, mark an angle $$55^{\circ}$$ using the protractor and name it as $$X$$. Now join $$AX$$.

Step 4: With $$B$$ as centre, mark an angle $$60^{\circ}$$ using the protractor and name it as $$Y$$. Now join $$BY$$.

Step 5: Mark the intersecting point of $$AX$$ and $$BY$$ as $$C$$.

Step 6: Draw the angle bisectors of any two angles of the triangle $$A$$ and $$B$$ (say).

Step 7: Mark the intersecting point of the angle bisectors as $$I$$, which is the incentre of the given triangle.

Step 8: Draw a perpendicular bisector from $$I$$ to one of the sides of the triangle $$AB$$ (say) to meet $$AB$$ at $$D$$.

Step 9: Measure the length of $$ID$$ to get the inradius of the incircle.

Step 10: With $$I$$ as centre and $$ID$$ as radius draw the incircle of the triangle. 