Vector Proofs: A Quadrilateral thing! (1 Viewer)

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Vector Proofs: A Quadrilateral thing #2!

Thanks lightgrav!
 
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forevergone said:
I need some help trying to prove that if the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram.
I've been attacking this problem for hours but its no good :\.
We had that problem on a geometry test in 9th grade, I will try digging it up and show you how :smile: ...if I can find it of course :rolleyes:
 
Any help is always appreciated!
 

lightgrav

Homework Helper
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you're given that dz = zb and az = zc , as your starting point.

What sums and differences of these equations show what you want?
 
az + zb = ab
cz + zd = cd

but az = zb, cz = cd therefore ab = cd!

Bah! That took like 5 minutes to see when I was spending 5 hours worth of time on it.

Thanks!
 

lightgrav

Homework Helper
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The key to this stuff is writing in symbols
JUST WHAT they tell you in words.

That's why everybody calls these things "Word Problems"!
 
But a new problem arises :\.
 

Diane_

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One way to do this is to show that you have a pair of congruent triangles. (There are actually several pair, but you only need one.) Remember the definition of a parallelogram - that'll give you the angles. There's one more property of parallelograms that will give you the sides that you need.
 
Diane_ said:
One way to do this is to show that you have a pair of congruent triangles. (There are actually several pair, but you only need one.) Remember the definition of a parallelogram - that'll give you the angles. There's one more property of parallelograms that will give you the sides that you need.
I need to do this through vector proofs, though. If I could use congruent triangles, I would've been long done this problem :).
 

daniel_i_l

Gold Member
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You just have to show that 1/2(dc+da) = 1/2db. that means that the middle of db touches the middle of ac. This is easy to prove. Start with the two equations:
db = da + ab
db = dc + cb
and try to solve for 1/2(dc+da) in terms of db.
 

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