Thread: The uniqueness of Euclid's propositions 10-12, book I

I have already shown how:
1) to bisect an angle,
2)bisect a line segment,
3)construct a perpendicular line at a given point on the line,
4)and how to construct a perpendicular line from a point not on the line,

all based on Euclid's propositions 9-12 in book I. I am then asked to prove:

5)the bisector in 1 is unique
6) the bisector in 2 is unique
7) the perpendicular line in 3 is unique
8) the perpendicular line in 4 is unique

it also hints that these proofs are usually done by contradiction. now i have done number five, but i am stuck at the other three. how are they unique and how can i prove them? because i can easily create another line that bisects the original line segment, but just at a different line segment. and for 7 and 8, i can create other lines as well that bisect the are perpendicular. any help would be great. my prof barely touched on uniqueness at all and now i have to construct all of this.

"Uniqueness" is, by its nature, a "negative"- there does not exist another object with the same properties. And, typically, negative statements are most simply proved by "contradiction".

To prove (6), that the point bisecting a line is unique (of course, "uniqueness" here refers to the uniqueness of the object, not the method of construction), assume there are two points, p1 and p2, that both bisect the line segment. Assume that p1 is, of the two, closer to one endpoint of the segment. That means, of course, that p2 is closer to the other endpoint. Now use the fact that these are both "bisectors" to show a contradiction.