Euclidean Geometry - Demonstration Exercise

[Samuel Gomes]

(a) Let be m a line and

the only two semiplans determined by m.

(i) Show that: If

are points that do not belong to

such

, so

and

are in opposite sides of m.

(ii) In the same conditions of the last item, show:

and

.

(iii) Determine the union result

, carefully justifying your answer.

(b) Let be

and

4 distincts points in a line

such

and

. Show

and

.

(c) Let be

distincts points in a line m such

. Under these conditions, show 2 distinct segments such the Union of both segments be equal to

, carefully justifying your answer.

Thanks for the help ^^

 

[HallsofIvy]

Proofs in Geometry and mathematics in general are heavily dependent on precise definitions! First, I assume this is in $R^2$ since it is not true in higher dimensions. But how are you defining "opposite sides" of a line? Is "PmA" the plane determined by the line m and the point A? If so HOW is a plane determined by a line and a point not on that line? For points A, B, and C, does "A-B-C" mean that B lies between A and C? If so, look at the segments AB and BC. What is their union?