1. **Problem Statement:** (i) Show that any two angle bisectors of a triangle meet in a point. (ii) Show that the three angle bisectors of a triangle meet in a point. 2. **Key Concept:** An angle bisector of a triangle is a line segment that divides an angle into two equal parts and extends from a vertex to the opposite side. 3. **Step (i): Two angle bisectors meet in a point** - Consider triangle ABC with angle bisectors from vertices A and B. - Let the angle bisector from A meet side BC at point D, and the angle bisector from B meet side AC at point E. - By the Angle Bisector Theorem, these bisectors divide the opposite sides proportionally. - The two bisectors intersect at a point I inside the triangle because each bisector is a line segment inside the triangle. 4. **Step (ii): Three angle bisectors meet in a point (Incenter)** - Let the third angle bisector from vertex C meet side AB at point F. - We need to show that the three bisectors intersect at a single point I. - Since I lies on the bisector of angle A and angle B, it is equidistant from sides AB and AC, and from sides AB and BC respectively. - By definition, the point equidistant from all sides of a triangle is the incenter. - Therefore, the third bisector from C also passes through I, proving concurrency. 5. **Summary:** - Any two angle bisectors intersect at a point inside the triangle. - The third angle bisector also passes through this point, so all three bisectors are concurrent. - This point is called the incenter, the center of the inscribed circle of the triangle. This completes the proof that the three angle bisectors of a triangle meet in a single point.