Proof:

Let's consider triangle ABC and triangle PQR.

Key Points:
1. The lines drawn parallel to the sides of triangle ABC create a smaller triangle PQR inside triangle ABC.
2. The sides of triangle PQR are parallel to the corresponding sides of triangle ABC.
3. The sides of triangle PQR and triangle ABC are proportional to each other.

Proof by Proportions:

Let's denote the lengths of the sides of triangle ABC as a, b, and c, where a is the length of side BC, b is the length of side AC, and c is the length of side AB.

The sides of triangle PQR are parallel to the corresponding sides of triangle ABC. Therefore, we can write the following proportions:

PQ || BC => PQ/BC = QR/AC
QR || AC => QR/AC = RP/AB
RP || AB => RP/AB = PQ/BC

By transitive property, we can write:

PQ/BC = QR/AC = RP/AB = k (a constant)

Proportionality of Perimeters:

Let's denote the lengths of the sides of triangle PQR as p, q, and r, where p is the length of side PQ, q is the length of side QR, and r is the length of side RP.

We know that the sides of triangle PQR and triangle ABC are proportional to each other. Therefore, we can write the following proportions:

p/a = q/b = r/c = k

Multiplying both sides of the proportion by a, b, and c respectively, we get:

p = ka
q = kb
r = kc

Perimeter of Triangle ABC:

The perimeter of triangle ABC is given by the sum of the lengths of its sides:

Perimeter(ABC) = a + b + c

Perimeter of Triangle PQR:

The perimeter of triangle PQR is given by the sum of the lengths of its sides:

Perimeter(PQR) = p + q + r

Substituting the values of p, q, and r from the proportionality equations, we get:

Perimeter(PQR) = ka + kb + kc

Factoring out k, we get:

Perimeter(PQR) = k(a + b + c)

Since a + b + c is the perimeter of triangle ABC, we can conclude that:

Perimeter(PQR) = k * Perimeter(ABC)

But we know that PQ/BC = QR/AC = RP/AB = k, so k = 1.

Therefore, Perimeter(PQR) = Perimeter(ABC)

Conclusion:

The perimeter of triangle PQR is equal to the perimeter of triangle ABC. Therefore, the perimeter of triangle PQR is double the perimeter of triangle ABC.