BE and CF are angle bisectors of triangle ABC that meet at I as shown below, and we have CE = 4, AE = 6, and AB = 8. Find BF
+12531 BE and CF are angle bisectors of triangle ABC that meet at I as shown below, and we have CE = 4, AE = 6, and AB = 8. Find BF
+129840 By Euclid Book VI Proposition 3, we have that
AF/FB = AC/ CB → x/ [8-x] = 10 / CB (1)
Also
EC / EA = BC / AB → 4/6 = BC / 8 → BC = 32/6 = 16/3 (2)
Substituting (2) into (1), we have that
x / [8 - x] = 10/[16/3] cross-multiply
16x/3 = 10 [8 -x]
16x = 30 [ 8 - x ]
16x = 240 - 30x
46x = 240
x = 120/23 = AF
8 - x = BF = 8 - 120/23 = 64/23
