Question

in the diagram, p1p2 and q1q2 are the perpendicular bisectors of ab and bc, respectively. a1a2 and b1b2 are the angle bisectors of ∠a and ∠b, respectively. the center of the inscribed circle of △abc is point. and the center of the circumscribed circle of △abc is point.

Explanation:

Step1: Recall circle - related definitions

The in - center of a triangle is the point of intersection of the angle bisectors of the triangle. The circum - center of a triangle is the point of intersection of the perpendicular bisectors of the sides of the triangle.

Step2: Identify the in - center

In \(\triangle ABC\), the angle bisectors \(A_1A_2\) and \(B_1B_2\) of \(\angle A\) and \(\angle B\) (and the third angle bisector not shown) intersect at the in - center. From the diagram, the in - center is point \(P\).

Step3: Identify the circum - center

The perpendicular bisectors \(P_1P_2\) of \(AB\) and \(Q_1Q_2\) of \(BC\) (and the third perpendicular bisector not shown) intersect at the circum - center. From the diagram, the circum - center is point \(R\).

Answer:

The center of the inscribed circle of \(\triangle ABC\) is point \(P\), and the center of the circumscribed circle of \(\triangle ABC\) is point \(R\)