In the known triangle ABC, the bisector of angle ABC and the bisector of angle ACB intersect at point F, make DF \ \ BC through point F, intersect AB at point D, intersect AC at point E Then: (1) how many isosceles triangles are there? Why? (2) what is the relationship between BD, CE and de? Please prove

In the known triangle ABC, the bisector of angle ABC and the bisector of angle ACB intersect at point F, make DF \ \ BC through point F, intersect AB at point D, intersect AC at point E Then: (1) how many isosceles triangles are there? Why? (2) what is the relationship between BD, CE and de? Please prove

1. There are two
DF / / BC, so angle DFB = angle CBF, and because BF bisects angle ABC, so angle DBF = angle CBF, so angle DBF = angle DFB, so DB = DF, so triangle DBF is isosceles triangle;
Let G be a point on the extension line of BC, DF / / BC, so angle DFC = angle FCG, and because CF bisects angle ACG, so angle ECF = angle FCG, so angle ECF = angle EFC, so EF = EC, so triangle ECF is isosceles triangle;
2. EF = EC, DF = de + EF, so DF = de + CE, because DB = DF, so BD = de + CE