Given a line l, let A be a point on l, and B another point not on l.
a. prove all the points of the line segment AB, except for A lie on the same side as B.
two lines can intersect in at most one point.
So, if there is a point P on AB on the opposite side from B, then PB intersects line l in some point besides A.
To prove that all the points of the line segment AB, except for A, lie on the same side as B, we need to use the concept of collinearity.
First, let's define a line segment AB as the collection of all points from the point A to the point B, where A and B are distinct points.
To prove that all the points of the line segment AB, except for A, lie on the same side as B, we can use a contradiction argument. We assume that there exists a point C on the line AB such that it lies on the opposite side of point B.
Assume that the line AB separates the plane into two parts: one side contains point A, and the other side contains point B. Now, we want to prove that the line segment AB is entirely contained on the side of A.
Consider a point C on the line AB. Since the line AB connects points A and B, point C must be expressed as C = (1 - t)A + tB, where t is a scalar value between 0 and 1.
If point C lies on the same side as A, then it should satisfy the following condition: C = (1 - t)A + tB lies on the same side as A. Let's call this condition (1).
Now, let's assume that point C lies on the opposite side of B. In this case, there exists a scalar value t' between 0 and 1 such that C' = (1 - t')A + t'B lies on the opposite side as B. Let's call this condition (2).
Now, if we take the difference of (1) and (2), we have:
(1 - t)A + tB - ((1 - t')A + t'B)
= A - tA + t'B - t'A - t'B
= (t' - t)(B - A)
Since (t' - t) is a scalar quantity, and (B - A) is a vector joining A and B, the result of the difference [(t' - t)(B - A)] is a vector. Let's call this vector R.
Now, if point C lies on the opposite side of B, then the vector R pointing from (1 - t')A + t'B to (1 - t)A + tB should point in the opposite direction of the vector pointing from B to A.
However, this contradicts the definition of a line segment, where all the points on the line segment AB travel in the same direction from A to B.
Therefore, our assumption that there exists a point C on the line AB that lies on the opposite side of B is false. Hence, all points of the line segment AB, except for A, lie on the same side as B.
QED.