a. 2AB = AM + AN 
=> 2AB = AM + AC + CN 
=> 2AB = AM + AB + CN 
=> AB = AM + CN 
=> AM + BM = AM + CN 
=> BM = CN 

b. BC cat MN tai F 
ve~ NE // BC ( E thuoc AB keo dai ) 
suy ra gocABC = gocAEN 
gocANE = gocACB 
ma gocABC = gocACB ( tam giac ABC can tai A ) 
=> hinh thang BCNE la hinh thang can 
=> CN = BE 
ma CN = BM ( cm cau a ) 
=> BM = BE 
BF // NE 
=> BF la duong trung binh tam giac MNE => MF = FN 
c. Xet tam giac KMN co 
KM vuong goc MN tai F 
MF = FN 
=> tam giac KMN can tai K 
=> MK = NK 
lai co KB = KC ( K thuoc phan giac goc BAC ) 
BM = CN ( cm cau a ) 
=> tam giac BKM = tam giac CKN (c.c.c) 
=> gocKCN = gocKBM ( = gocABK ) 

gocABC=gocACB(tam giac ABC can) 
gocKBC=gocKCB(tam giac KBC can) 
=> gocABC + gocKBC = gocACB + gocKCB 
=> gocABK = gocACK 
ma gocABK = gocKCN 
=> gocKCN = gocACK 
ma gocKCN + gocACK = 180* 
=> gocKCN = 90* => KC vuong goc AN