Given:
- Triangle ABC
- AD is the median on BC

To prove:
The area of triangle ABD is equal to the area of triangle ACD.

Proof:
Let's begin by drawing the given triangle ABC and its median AD.

Step 1: Drawing the Triangle and Median
- Draw a triangle ABC on a piece of paper.
- Label the vertices as A, B, and C.
- Draw a line segment AD from vertex A to the midpoint D of side BC.

Step 2: Understanding the Problem
- To prove that the area of triangle ABD is equal to the area of triangle ACD, we need to use the concept of triangles with the same base and height having equal areas.
- In this case, we will consider the base as AD and the height as the perpendicular distance from vertex B and vertex C to the base AD.

Step 3: Proving the Triangles have the Same Base
- Since D is the midpoint of BC, we can conclude that AD is the median, which means it divides BC into two equal parts.
- This implies that BD = CD, so both triangle ABD and triangle ACD have AD as their common base.

Step 4: Proving the Triangles have the Same Height
- To establish that the triangles have the same height, we need to prove that the perpendicular distance from vertex B to AD is equal to the perpendicular distance from vertex C to AD.
- Let's denote the perpendicular distance from vertex B to AD as h1 and the perpendicular distance from vertex C to AD as h2.

Step 5: Establishing the Heights are Equal
- Since AD is the median, it divides the side BC into two equal parts, BD and CD.
- As a result, the perpendicular distance from vertex B to AD is equal to the perpendicular distance from vertex C to AD.
- Therefore, h1 = h2.

Step 6: Applying the Formula for Triangle Area
- Now that we have established that both triangles have the same base (AD) and the same height (h1 = h2), we can use the formula for the area of a triangle to prove our statement.
- The formula for the area of a triangle is: Area = (base * height) / 2.

Step 7: Calculating the Areas of the Triangles
- Applying the formula for triangle area, we can calculate the area of triangle ABD as (AD * h1) / 2.
- Similarly, the area of triangle ACD can be calculated as (AD * h2) / 2.

Step 8: Comparing the Areas
- Since h1 = h2, we have (AD * h1) / 2 = (AD * h2) / 2.
- This implies that the area of triangle ABD is equal to the area of triangle ACD.

Step 9: Conclusion
- Therefore, we have successfully proved that the area of triangle ABD is equal to the area of triangle ACD using the concept of triangles with the same base and height having equal areas.

Summary:
- Triangle ABC is given with AD as the median on BC.
- We proved that the area