Question

In the diagram, P1P2 and Q1Q2 are the perpendicular bisectors of line AB and BC , respectively. A1A2 and B1B2 are the angle bisectors of angle A and B , respectively. The center of the inscribed circle of ΔABC is point ____ , and the center of the circumscribed circle of ΔABC is point ____ .

Answer 1

I hope I can show the image, but I hope this may help you, the answer is:

The center of the inscribed circle of ΔABC is point S , and the center of the circumscribed circle of ΔABC is point P.

Answer 2

Answer:  The complete statement is

The center of the inscribed circle of ΔABC is point S , and the center of the circumscribed circle of ΔABC is point P .

Step-by-step explanation:  Given that $P_1P_2$ and $Q_1Q_2$ are the perpendicular bisectors of the lines AB and BC respectively.

And, $A_1A_2$ and $B_1B_2$ are the angle bisectors of the angles A and B respectively.

We are to find the centre of the inscribed circle of ΔABC and the centre of the circumscribed circle of ΔABC.

INCENTRE : The point at which the angle bisectors of the three angles of a triangle meet is called the INCENTRE of the triangle. This point is the centre of the inscribed circle of the triangle.

The angle bisectors of angles A and B, that is $A_1A_2$ and $B_1B_2$  meet at the point S.

Also, since the three angle bisectors are concurrent, so the third angle bisector must also pass through S.

Thus, the point S is the incentre of ΔABC and so, the centre of the inscribed circle of ΔABC is point S.

CIRCUMCENTRE : The point at which the perpendicular bisectors of the three sides of a triangle meet is called the CIRCUMCENTRE of the triangle. This point is the centre of the circumscribed circle of the triangle.

The perpendicular bisectors of sides AB and BC, that is $P_1P_2$ and $Q_1Q_2$  meet at the point P.

Also, since the three perpendicular bisectors are concurrent, so the third perpendicular bisector must also pass through P.

Thus, the point P is the circumcentre of ΔABC and so, the centre of the circumscribed circle of ΔABC is point P.

Hence, the complete statement is

The center of the inscribed circle of ΔABC is point S , and the center of the circumscribed circle of ΔABC is point P .