A Quadrilateral Theorem

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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?

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Posted · Report post

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

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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?

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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

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Posted · Report post

here is the figure of what i am talking about

post-53485-0-97784700-1365728351_thumb.p

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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.

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