A Quadrilateral Theorem

[BMAD]

Given four points A, B, C, D, and four directly similar quadrilaterals AP₁P₂B, CQ₁Q₂B, CR₁R₂D, AS₁S₂D with respective centroids P, Q, R, S, let K, L, M, and N be the midpoints of the segments P₁Q₂, Q₁R₂, R₁S₂ and S₁P₂ respectively (or of P₁S₂, S₁R₂, R₁Q₂ and Q₁P₂), and let V, W, X be the centroids of the quadrilaterals ABCD, PQRS, KLMN respectively.

Then what is special about PQRS, KLMN, and point W?

[BMAD]

PQRS is a parallelogram....can you tell me why?

[bonanova]

Given four points A, B, C, D, and four directly similar quadrilaterals AP₁P₂B,

I'm struggling with picturing these similar quadrilaterals.

Proportional sides are needed, No?

If AB is common, which sides are proportional?

[BMAD]

There are 5 ways to prove a Quadrilateral is a Parallelogram.

-Prove both pairs of opposite sides congruent

-Prove both pairs of opposite sides parallel

-Prove one pair of opposite sides both congruent and parallel

-Prove both pairs of opposite angles are congruent

-Prove that the diagonals bisect each other

[BMAD]

here is the figure of what i am talking about

[BMAD]

Given a quadrilateral ABCD with equilateral triangles ABP, BCQ, CDR and DAS constructed on the sides so that say the first and third are exterior to the quadrilateral, while the second and the fourth are interior to the quadrilateral, prove that quadrilateral PQRS is a parallelogram.