Proof:

Let's consider triangle ABC with AD as the median and AM perpendicular to BC.

1. Applying the Pythagorean theorem:
In triangle ABC, we can apply the Pythagorean theorem to obtain the following equations:
AC^2 = AB^2 + BC^2 ...(1)
AD^2 = AB^2 + BD^2 ...(2)

2. Proving triangles ABD and AMC are similar:
Since AM is perpendicular to BC, triangles ABD and AMC are similar by the AA similarity criterion.
This means that the corresponding angles in both triangles are equal, and we can write:
∠DAB = ∠CAM ...(3)
∠ADB = ∠ACM ...(4)

3. Using the property of similar triangles:
From the property of similar triangles, we know that the ratio of the lengths of corresponding sides in similar triangles is equal. In this case, we have:
AB/AD = AC/AM ...(5)
AB/AD = BC/DM ...(6)

4. Solving equations (5) and (6):
From equations (5) and (6), we can equate the ratios and solve for AC in terms of AD, BC, and DM:
AC/AM = BC/DM
AC = (BC/DM) * AM ...(7)

5. Using the property of medians in a triangle:
In a triangle, the median divides the opposite side into two equal segments. Therefore, we have:
BD = CD ...(8)

6. Substituting values in equation (2):
Substituting equation (8) into equation (2), we get:
AD^2 = AB^2 + CD^2 ...(9)

7. Substituting values in equation (7):
Substituting equations (3) and (4) into equation (7), we get:
AC = (BC/DM) * AD ...(10)

8. Simplifying equation (10):
Multiplying both sides of equation (10) by AD, we get:
AC * AD = BC * DM ...(11)

9. Substituting values in equation (1):
Substituting equation (11) into equation (1), we get:
AC^2 = AD^2 + BC * DM ...(12)

10. Simplifying equation (12):
Rearranging equation (12), we have:
AC^2 - AD^2 = BC * DM ...(13)

11. Proving equation (13):
From equation (13), we can see that AC^2 - AD^2 is equal to BC * DM. Therefore, the given statement is proved.

So, we have proven that in triangle ABC, with AD as the median and AM perpendicular to BC, AC^2 = AD^2 + BC * DM.