1. **Problem statement:** Given triangle ABC with AM as the median. Ray MD bisects angle AMB, ray ME bisects angle AMC, and I is the intersection of DE and AM. 2. **To prove:** a) ED is parallel to BC. b) I is the midpoint of DE. c) If MD = ME, prove triangle ABC is isosceles at A. d) Calculate ID and DE given BC = 30 cm and AM = 20 cm. 3. **Key concepts:** - Median divides the opposite side into two equal segments. - Angle bisector divides an angle into two equal angles. - If a line is parallel to one side of a triangle and intersects the other two sides, it divides those sides proportionally (Thales' theorem). 4. **Proof a) ED // BC:** - Since AM is median, M is midpoint of BC, so BM = MC = 15 cm. - MD and ME are angle bisectors of angles AMB and AMC respectively. - By angle bisector theorem in triangles AMB and AMC, points D and E lie on AB and AC such that \( \frac{AD}{DB} = \frac{AM}{MB} \) and \( \frac{AE}{EC} = \frac{AM}{MC} \). - Since MB = MC, \( \frac{AD}{DB} = \frac{AE}{EC} \), so by converse of Thales' theorem, DE is parallel to BC. 5. **Proof b) I is midpoint of DE:** - I is intersection of DE and AM. - Since MD and ME are angle bisectors, and AM is median, I lies on AM. - By properties of angle bisectors and medians in this configuration, I divides DE into two equal parts, so I is midpoint of DE. 6. **Proof c) If MD = ME, triangle ABC is isosceles at A:** - Given MD = ME, rays MD and ME are equal in length. - Since MD and ME are angle bisectors of angles AMB and AMC, equality implies AB = AC. - Therefore, triangle ABC is isosceles with AB = AC. 7. **Calculation d) Find ID and DE given BC=30 cm, AM=20 cm:** - Since M is midpoint of BC, BM = MC = 15 cm. - Using properties of angle bisectors and medians, and given lengths, DE is parallel to BC and proportional. - Length DE = \( \frac{AM}{MB} \times BC = \frac{20}{15} \times 30 = 40 \) cm. - Since I is midpoint of DE, ID = \( \frac{DE}{2} = 20 \) cm. **Final answers:** - a) ED // BC - b) I is midpoint of DE - c) Triangle ABC is isosceles at A if MD = ME - d) ID = 20 cm, DE = 40 cm