Vector Proofs: A Quadrilateral thing! (1 Viewer)

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forevergone

Vector Proofs: A Quadrilateral thing #2!

Thanks lightgrav!

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celticsthree4

re:

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 ...if I can find it of course

forevergone

Any help is always appreciated!

lightgrav

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

What sums and differences of these equations show what you want?

forevergone

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
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"!

forevergone

But a new problem arises :\.

Diane_

Homework Helper
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.

forevergone

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
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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