VECTOR GEOMETRY AND BASIC PROPERTIES OF PARAELLOGRAMSNOTE. The following should be inserted after the proof of Proposition III.3.6 on page107 of the file geometrynotes3a.pdf.In the notes there are synthetic proofs for several basic theorems about parallelograms.Our purpose here is to give some alternate proofs using vectors.REVIEW. By Exercise I.4.3 in the file math133exercises1.pdf, if we are given noncollinearpoints a, b and d in R2and c = b + d − a, then the four points a, b, c and d (inthat order) form the vertices of a parallelogram.This observation has two simple but important consequences:b − a = c − d , c − b = d − aThese follow directly from the formula for c given above, and they immediately imply allthe basic measurement properties of a parallelogram:If a, b, c and d (in that order) form the vertices of a parallelogram, then the followinghold:(i) d(a, b) = d(c, d)(ii) d(a, d) = d(b, c)(iii) |6dab| = |6bcd|(iv) |6dab| + |6adc| = 180◦Derivations. Since b − a = c − d and c − b = d − a, we have the following:d(a, b) = |b − a| = |c − d| = d(c, d)d(a, d) = |d − a| = |b − c| = d(b, c)cos |6dab| =(d − a) · (b − a)|d − a| · |b − a|=(b − c) · (d − c)|b − c| · |d − c|= cos |6bcd|Since the cosine function is a strictly decreasing function on angle measurements between0 and 180 degrees, the third line implies (iii). Of course, the first two lines imply (i)and (ii). Finally, to prove (iv) we proceed much as in (iii), but the outcome is slightlydifferent:cos |6adc| =(a − d) · (c − d)|a − d| · |c − b|= −(a − d) · (c − d)−|a − d| · |c − b|=−(d − a) · (b − a)|d − a| · |b − a|= − cos |6dab|1Since cos θ = − cos ϕ if and only if θ and ϕ are supplementary, the conclusion of (iv)follows immediately.Here is one more standard result on parallelograms:If a, b, c and d (in that order) form the vertices of a parallelogram, then the diagonallines ac and bd intersect at a point which is the midpoint of both [ac] and [bd] (in words,the diagonals of the parallelogram bisect each other).To prove this it is only necessary to check that12(a + c) and12(b + d) are the samepoint. This can be done by noting that the formula for c implies12(a + c) =12a + (b + d − a)=12(b + d)
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