Vector Proofs: A Quadrilateral thing

[forevergone]

Vector Proofs: A Quadrilateral thing #2!

Thanks lightgrav!

 

[celticsthree4]

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]

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]

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_]

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]

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]

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.