Proof of the Formula for Area of Similar Triangles

Introduction:
In this proof, we will show that the ratio of the area of two similar triangles is equal to the square of the ratio of their corresponding sides. We will use the given conditions that triangle ABC is similar to triangle PQR, and ad and ps are bisectors of their corresponding angles a and p, respectively.

Proof:
Let us denote the sides of the two triangles as follows:
AB = a, BC = b, AC = c (of triangle ABC)
PQ = p, QR = q, PR = r (of triangle PQR)

Step 1: Prove that the ratio of the lengths of corresponding sides is equal to the ratio of the lengths of the bisectors of their corresponding angles.
From the given conditions, we know that triangle ABC is similar to triangle PQR. Therefore, we can write:
a/p = b/q = c/r
Now, let us consider the bisectors of the corresponding angles. In triangle ABC, ad is the bisector of angle A, and in triangle PQR, ps is the bisector of angle P. By the angle bisector theorem, we know that:
ad/b = c/q
ps/r = p/q
Therefore, we have:
ad/ps = (ad/b) / (ps/r) = (c/q) / (p/q) = c/p
Thus, we have proven that ad/ps = c/p.

Step 2: Prove that the ratio of the areas of the two triangles is equal to the square of the ratio of the lengths of their corresponding sides.
The area of triangle ABC can be written as:
Area(ABC) = (1/2) * a * b * sin(A)
Similarly, the area of triangle PQR can be written as:
Area(PQR) = (1/2) * p * q * sin(P)
Since the two triangles are similar, we have:
sin(A) = sin(P) = k (some constant)
Therefore, we can write:
Area(ABC) / Area(PQR) = (a * b) / (p * q)
Using the ratio of corresponding sides that we proved in Step 1, we have:
Area(ABC) / Area(PQR) = (a/p) * (b/q) = (c/r) * (b/q) = (c/p) * (b/r) = (ad/ps) * (bd/qr)
We know that ad/ps = c/p, and we also know that bd/qr = c/p (since triangle ABC and PQR are similar). Therefore, we can simplify the above equation as:
Area(ABC) / Area(PQR) = (c/p) * (c/p) = (c/p)^2
Substituting c/p with ad/ps (from Step 1), we get:
Area(ABC) / Area(PQR) = (ad/ps)^2
Thus, we have proven that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Conclusion:
In this proof, we showed that the ratio of the area of two similar triangles is equal to the square of the ratio of their corresponding sides, using the given conditions that triangle ABC is similar to triangle PQR