Proof:

Let's consider the given information step by step to prove that FD = 1/3 FE.

1. Construction:
Draw the line CE such that CE = 1/2 AC.
Construct the line DP parallel to AB, intersecting CE at point P.
Construct the line DQ parallel to AB, intersecting CE at point Q.
Join the points F and D.

2. Triangles and their Properties:
Let's denote the lengths of segments as follows:
AC = a
CE = 1/2AC = 1/2a
CD = DB = a/2 (as D is the midpoint of BC)

3. Using Similarity:
Triangles ADC and ADF are similar because they have:
- Angle ADC = Angle ADF (both right angles)
- Angle DAC = Angle DAF (common angle)
Therefore, we have:
AD/AD = AC/AF
1 = a/AF
AF = a

Triangles CEP and CFD are similar because they have:
- Angle CEP = Angle CFD (both right angles)
- Angle CPE = Angle CDF (common angle)
Therefore, we have:
CE/CF = CP/CD
1/2a/CF = CP/a/2
CF = CP/4

4. Using Proportions:
In triangle CFP, using the property of similar triangles, we can write:
CP/CF = CD/CP
CP^2 = CF*CD
CP^2 = (CP/4)*(a/2)
CP = a/4

In triangle CDE, using the property of similar triangles, we can write:
CE/CD = CF/CE
1/2a/a/2 = CF/(1/2a)
CF = 1/4a

5. Relation between FD and FE:
In triangle AEF, using the property of similar triangles, we can write:
AF/FE = AD/DE
a/FE = a/(DE + EF)
1/FE = 1/(DE + EF)
FE = DE + EF

In triangle CDE, we have:
CD = a/2
DE = CE - CD
DE = 1/2a - a/2
DE = (1 - a)/2a

Substituting the values of DE and EF in the equation FE = DE + EF, we get:
FE = (1 - a)/2a + EF

6. Final Calculation:
Since CP is parallel to AB, we have:
DE = DP = (1 - a)/2a

Substituting the value of DE in the equation for FE, we get:
FE = (1 - a)/2a + EF

Since DP is parallel to AB, triangles CDP and CFD are similar. Therefore, we have:
CP/CF = CD/CD
CP/CF = 1
CP = CF

Substituting the value of CP in the equation for FE, we get:
FE = (1 - a)/2a + CF

Since CP is parallel to AB, triangles CFP and CED are similar. Therefore, we have:
CP/CF = CE/CD
CP/CF = 1/2
CP = CF/2

Substituting the value