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**1. Homework Statement**

23. In a ABCD quadrilateral let P,Q,R,S be midpoints of sides AB,BC,CD and DA. Let X be the intersection of BR and DQ, and let Y be the intersection of BS and DP. If ##\vec{BX}=\vec{YD} ## show that ABCD is a parallelogram .

**2. Homework Equations**

## (\vec{a}\cdot\vec{b})=0## then a and b are perpendicular

I can't think of any other useful ones

**3. The Attempt at a Solution**

So when I first saw this problem I didn't really have an idea how to solve it so I just tried figuring out what are the rations that X divides BR into. I calculated that the ratio is 2/3. I did the same for Y and DP and also got the ratios 2/3. Which I though was good since I figured out that ##\vec{BX}=\vec{YD} ## really are the same. However I still didn't have any idea how to continue so then I decided to figure out ratios of AY,AX,XY and found out that AY/(AC)=XY/(AC)=XC/(AC)=1/3 however I don't see how any of this would help me show that ABCD is a parallelogram. I'm kinda lost here.

I though about trying to show that ## (\vec{a}\cdot\vec{b})=/=0## since that would mean that a and b are not perpendicular and that we have a parallelogram however I was not able to find a way to show that.

Any help is greatly appreciated.