ABCD is a quadrilateral; P, Q, R and S are the points of trisection of sides AB, BC, CD and DA respectively and are adjacent to A and C; prove that PORS is a parallelogram.
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We have a quadrilateral ABCD. Points P, Q, R, and S are trisection points on sides AB, BC, CD, and DA respectively, positioned adjacent to vertices A and C. Our goal is to prove that quadrilateral PQRS forms a parallelogram.
We'll use vector methods to solve this problem. Let O be the origin, and let vectors a, b, c, d represent the position vectors of vertices A, B, C, D respectively. Since P, Q, R, S are trisection points, we can express their position vectors in terms of the vertex vectors. Point P divides AB in ratio one to two, so its position vector is two-thirds a plus one-third b.
Now we calculate the vectors representing opposite sides of quadrilateral PQRS. First, vector PQ equals q minus p. Substituting our expressions and simplifying, we get two-thirds times the vector from a to c. Similarly, vector SR equals r minus s, which also simplifies to two-thirds times the vector from a to c. Since these vectors are equal, sides PQ and SR are parallel and have equal length.
Let's verify the other pair of opposite sides. Vector PS equals s minus p, which simplifies to one-third times the vector from b to d. Similarly, vector QR equals r minus q, which also gives us one-third times the vector from b to d. Since both pairs of opposite sides are parallel and equal, we have proven that PQRS is indeed a parallelogram.
To summarize what we've learned: When we take trisection points on the sides of any quadrilateral, these points always form a parallelogram. Vector methods provide an elegant way to prove such geometric properties. The key insight is that opposite sides being parallel and equal is the defining characteristic of a parallelogram.