Well, two things first,

1) HAPPY NEW YEAR TO ALL !

2) My Pittsburgh Steelers have just eliminated the Cinci Bungles from NFL playoffs! WE DEY!

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Okay, for your Question #1 now.

Here is one way. It's very crude because I'd be using numbers or definite lengths for the vectors. Variables are better for proofs, but hey, it's holiday today. Why crack our heads on a holiday?

Say we have triangle whose vertices are A(0,0), B(8,6) and C(10,0).

The two sides AB and AC are halved each. D(4,3) is midpoint of AB. E(5,0) is midpoint of AC.

b) Is DE half as long as BC?

DE = sqrt[(4-5)^2 +(3-0)^2] = sqrt[1 +9] = sqrt(10) units long.

BC = sqrt[(8-10)^2 +(6-0)^2] = sqrt[4 +36] = sqrt(40) = 2sqrt(10) units long.

Therefore, yes, DE is half in length of BC.-------proven.

a) Is DE parallel to BC?

DE = AE -AD -----in vectors.

DE = ((4 -5),(3-0))

DE = (-1,3) ---------------***

BC = AC -AB .....in vectors.

BC = ((8-10),(6-0))

BC = (-2,6)

or,

BC = 2(-1,3) ---------------***

Since BC is just DE multiplied by a scalar of 2, then BC and DE are parallel. ----proven.

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That's all for now. I just wanted to say Happy New Year and to needle the Bungleds.