: Cho tam giác đều ABC. Trên tia đối diện của BA và CA lấy các điểm M và N sao cho . Gọi I là giao điểm của MC và BN. a) Chứng minh: MI = NI. b) Phân giác của góc AMC cắt AI và AN theo thứ tự O và K. BO cắt AN tại Q. CMR: OKQ là tam giác cân.
1 trả lời
65 Tham khảo câu trả lời từ Trợ lý ảo Lazi a) Ta có tam giác ABC đều nên AM = AN. Gọi G là giao điểm của BM và CN. Ta có MG // AC và NG // AB, suy ra MG = NG. Áp dụng định lí Menelaus cho tam giác ABC và đường thẳng song song với BC qua I, ta có: $\frac{BM}{MC} \cdot \frac{CI}{IA} \cdot \frac{AN}{NB} = 1$ $\frac{BM}{MC} = \frac{AN}{NB}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + NG}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + MG}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM + MG} = \frac{AN}{AN + BM}$ $\frac{BM}{BM +