Vector Proofs: A Quadrilateral thing #2!
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
Any help is always appreciated!
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.
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 :\.
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 :).
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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