**Proof:**

Given: In an equilateral triangle ABC, D is a point on side BC such that BD = 1/3 BC.

To prove: 9(AD)^2 = 7(AB)^2

**Construction:**

1. Draw an equilateral triangle ABC.
2. Mark the point D on side BC such that BD = 1/3 BC.

**Proof:**

We need to prove that 9(AD)^2 = 7(AB)^2.

To begin, let's determine the relationship between the sides of the triangle ABC.

- In an equilateral triangle, all sides are equal. Therefore, AB = BC = AC.

Now, let's consider the triangles ADC and ADB.

- Triangle ADC: In triangle ADC, AD is the common side, and AC = AB as AC = BC = AB.
- Triangle ADB: In triangle ADB, AD is the common side, and AB = AB (reflexive property).

Since AD is the common side in both triangles ADC and ADB, we can conclude that these two triangles are congruent by the Side-Side-Side (SSS) congruence criterion.

Therefore, we can say that triangle ADC is congruent to triangle ADB.

Now, let's use the property of congruent triangles to determine the relationship between their corresponding sides.

- In congruent triangles, corresponding sides are equal. Therefore, DC = DB.

Let's consider the ratio of the lengths of the sides DC and DB.

- DC = DB implies that DC/DB = 1.

Since BD = 1/3 BC, we can write BD/BC = 1/3.

- Comparing the ratios, we have DC/DB = 1 and BD/BC = 1/3.

Now, let's consider the ratio of the areas of triangles ADC and ADB.

- The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Therefore, (Area of ADC)/(Area of ADB) = (DC/DB)^2 = 1^2 = 1.

Since triangle ADC is congruent to triangle ADB, their areas are equal.

Hence, (Area of ADC) = (Area of ADB).

Now, let's calculate the areas of triangles ADC and ADB.

- The area of an equilateral triangle can be given by the formula (sqrt(3)/4) * (side)^2.

Let AB = BC = AC = a (let's assume the side length of the equilateral triangle as 'a').

- The area of ADC = (sqrt(3)/4) * (AC)^2 = (sqrt(3)/4) * a^2.
- The area of ADB = (sqrt(3)/4) * (AB)^2 = (sqrt(3)/4) * a^2.

Since the areas of triangles ADC and ADB are equal, we can write:

(sqrt(3)/4) * a^2 = (sqrt(3)/4) * a^2.

Now, let's simplify the equation:

(sqrt(3)/4) * a^2 = (sqrt(3)/4) * a^2.

This equation holds true for any value of 'a', which means it is an identity.

Now, let's substitute the value of BD in terms of BC into the equation to