Question

Can someone help prove this statement?

Let E be the center of AB in the triangle ABC. Furthermore, A is the center of DC, and line BD hits line through C and E in F.

Show that F shares the line DB in the 2: 1 ratio.

Answer 1

see explanation.

Explanation:

Draw a line `AG`, parallel to `CF`, as shown in the figure,
`=> DeltaBEF and DeltaBAG` are similar,
as `AE=EB, => GF=FB`,
`DeltaDAG and DeltaDCF` are similar,
similarly, as `DA=AC, => DG=GF`,
`=> DG=GF=FB or DG:GF:FB=1:1:1`,
`=> DF:FB=(DG+GF):FB=2:1`