Question

the internal bisector of angle C of triangle ABC cuts AB at F .the internal bisector of Angle B cuts CF at I . prove that AF / FI = AC / CI

Answer 1

All three angle bisectors of a triangle are concurrent, intersecting at the triangle’s incenter. This means AI bisects angle A. So basically, you are just trying to prove the angle bisector theorem as applied to triangle ACF.

Draw triangle ACF with the internal bisector of angle A cutting CF at I. Draw a line through C parallel to AF and extend AI so it cuts this line at G. Now angles FAI and CGI are congruent (alternate interior angles in parallel lines) and angles AIF and GIC are congruent (vertical angles) so triangles AIFand CIG are similar. Thus AF:FI = GC:CI.

Angle CAI is congruent to angle FAI (AI is an angle bisector) so angles CAI and CGI are also congruent making triangle ACG isosceles. Thus AC = GC and your desired result follows by substitution.