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