# 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.

Jan 16, 2018

see explanation.

#### Explanation:

Draw a line $A G$, parallel to $C F$, as shown in the figure,
$\implies \Delta B E F \mathmr{and} \Delta B A G$ are similar,
as $A E = E B , \implies G F = F B$,
$\Delta D A G \mathmr{and} \Delta D C F$ are similar,
similarly, as $D A = A C , \implies D G = G F$,
$\implies D G = G F = F B \mathmr{and} D G : G F : F B = 1 : 1 : 1$,
$\implies D F : F B = \left(D G + G F\right) : F B = 2 : 1$