Given:
- Triangle ABC
- D is the midpoint of side BC
- AD is bisected at point E
- BE produced cuts AC at point X

To Prove: BE:EX = 3:1

Proof:

1. Draw a diagram:

Let's start by drawing a diagram to visualize the given information.

2. Identify the given information:

From the question, we can identify the following:

- D is the midpoint of side BC
- AD is bisected at point E
- BE produced cuts AC at point X

3. Identify relevant triangles:

In order to prove the given statement, we need to identify relevant triangles that will help us establish a relationship between BE and EX.

Let's consider triangle ABE and triangle XDE.

4. Use midpoint theorem:

Since D is the midpoint of side BC, we can apply the midpoint theorem to triangle ABC to conclude that AD is parallel to EX and DE is parallel to BX.

5. Use alternate interior angles:

Using the fact that AD is parallel to EX, we can conclude that angle AED is equal to angle DEX (alternate interior angles).

6. Use vertically opposite angles:

We can also notice that angle XDE is equal to angle BAE (vertically opposite angles).

7. Use angle bisector theorem:

Since AD is bisected at point E, we can apply the angle bisector theorem to triangle ABD to conclude that BE:EX = AB:AX.

8. Use similar triangles:

By using the angles we established as equal in step 5 and step 6, we can prove that triangle ABE is similar to triangle XDE (angle-angle similarity).

9. Use corresponding sides:

Since triangle ABE is similar to triangle XDE, we can conclude that the ratio of corresponding sides is equal:

BE:EX = AB:AX = DE:DX

10. Use midpoint theorem again:

Since D is the midpoint of BC, we can apply the midpoint theorem to triangle ABC to conclude that AC is parallel to DX and AX is parallel to BC.

11. Use corresponding sides again:

Using the fact that AC is parallel to DX, we can conclude that AX:XC = AD:DX.

12. Substitute values:

Substituting the value of AD as 2 and DX as 1 (since AD is bisected at E), we get:

AX:XC = 2:1

13. Use corresponding sides one last time:

Using the fact that AX is parallel to BC, we can conclude that AB:AX = AC:XC.

14. Substitute values again:

Substituting the value of AX as 2 and XC as 1 (from step 12), we get:

AB:2 = AC:1

15. Use the given information:

Since D is the midpoint of BC, we can conclude that AB = AC.

16. Simplify the ratio:

Using the values from step 15, we can simplify the ratio obtained in step 14:

AB:2 = AB:1

17. Conclude the proof:

From step