Hello, bigb!
a) Let be a quadrilateral.
Consider the points: .
Prove that EFGH is a parallelogram.
I must assume that those equations refer to the lengths of line segments,
and that the sides are divided in the ratio 1:2 or 2:1.
The diagram looks somthing like this: Code:
A E B
o - - - - - o - - - - - o
* * * *
* * * *
* * * *
H o * *
* * * *
* * o F
* * * *
* * * *
* * * *
o - - - - - - - - - - - - - - o - - - - - - - - - - - o
D G C
Draw diagonal
Consider
. .
. .
. .
Hence: .
. . .[1]
. . .[2]
Consider
. .
. .
. .
Hence: .
. . .[3]
. . .[4]
From [1] and [3]: .
From [2] and [4]: .
Theorem: If two sides of a quadrilateral are equal and parallel,
. . . . . . . the quadrilateral is a parallelogram.
Therefore, is a parallelogram.