Incenter and incircles of a triangle  Geometry  Khan Academy  Summary and Q&A
TL;DR
Angle bisectors in triangles intersect at a unique point called the incenter, which is equidistant from the sides of the triangle and is the center of the incircle inscribed within the triangle.
Questions & Answers
Q: What are the properties of points on angle bisectors in triangles?
Points on angle bisectors in triangles are equidistant from the two sides of the angle they bisect. This can be proven by dropping perpendiculars from the point to the sides and showing that the distances are equal.
Q: How can we identify the incenter of a triangle?
The incenter of a triangle is the point of intersection of the angle bisectors. It is the unique point inside the triangle that is equidistant from the sides and sits on all three angle bisectors.
Q: What is the significance of the incenter and incircle in a triangle?
The incenter is important because it divides the angles of the triangle into congruent angles. The incircle with the incenter as its center is a circle that is tangent to all three sides of the triangle.
Q: How is the inradius of the incircle determined?
The inradius of the incircle is equal to the distance between the incenter and any of the sides of the triangle. It is also equal to the length of the perpendiculars dropped from the incenter to each side of the triangle.
Summary & Key Takeaways

Angle bisectors in triangles divide the angles into two congruent angles.

The point of intersection of the angle bisectors is called the incenter.

The incenter is equidistant from the sides of the triangle and is the center of the incircle inscribed within the triangle.