Proof:

To prove that 3AB^2 = 4AD^2 in an equilateral triangle ABC, where AD is the altitude drawn from A to side BC, we can use the Pythagorean theorem and the properties of an equilateral triangle.

Properties of an equilateral triangle:
- All sides of an equilateral triangle are equal.
- All angles of an equilateral triangle are equal and measure 60 degrees.
- The altitude of an equilateral triangle bisects the base.

Proof:

We start by drawing the equilateral triangle ABC and the altitude AD from A to side BC.

Step 1: Label the triangle and the given information.

Let's label the triangle as ABC and the altitude as AD.

Step 2: Use the property that the altitude of an equilateral triangle bisects the base.

Since AD is the altitude of the equilateral triangle ABC, it bisects the base BC at point E.

Step 3: Use the property that all angles of an equilateral triangle measure 60 degrees.

Since ABC is an equilateral triangle, angle BAC measures 60 degrees.

Step 4: Use the Pythagorean theorem.

In right triangle ADE, we can apply the Pythagorean theorem:

AD^2 + DE^2 = AE^2

Step 5: Determine the lengths of the sides and the altitude.

Since ABC is an equilateral triangle, all sides are equal, so AB = BC = AC.

Since AD is the altitude, it is perpendicular to BC, so angle ADE is a right angle.

Based on the properties of an equilateral triangle, we know that AD bisects BC at point E, so BE = CE = BC/2.

Step 6: Calculate the lengths of AE and DE.

Using the properties of an equilateral triangle, we can determine the lengths of AE and DE.

Since ABC is an equilateral triangle, triangle ABE is a 30-60-90 right triangle.

In a 30-60-90 triangle, the length of the hypotenuse is twice the length of the shorter leg.

Therefore, AE = 2 * DE.

Step 7: Substitute the lengths into the Pythagorean theorem equation.

Substituting the lengths into the equation AD^2 + DE^2 = AE^2, we have:

AD^2 + DE^2 = (2 * DE)^2

AD^2 + DE^2 = 4 * DE^2

Step 8: Simplify the equation.

Simplifying the equation, we have:

AD^2 = 3 * DE^2

Step 9: Substitute the lengths of DE and BC.

Since DE = BC/2, we can substitute this length into the equation:

AD^2 = 3 * (BC/2)^2

AD^2 = 3 * (BC^2/4)

AD^2 = 3 * BC^2/4

Step 10: Use the property that all sides of an equilateral triangle are equal.

Since AB = BC = AC, we can substitute AB for BC in the equation:

AD^2 = 3 * AB^