Theorem 2.7: Given points   A   and   B   and a line whose equation is   ax + by = c,   where   A   is either on the line or on the same side of the line as   B,   every point on the line segment between   A   and   B   is on the same side of the line as  B.

Proof: Let   A = (x1, y1)   and let   B = (x2, y2).

The equation of the line is   ax + by = c.   We can assume that

ax1 + by1 < c

because if we find the inequality going the other way, the proof will be similar. Since   A  is supposed to be either on the line or on the same side as   B,   we have

ax2+ by2 < c

Let   C   be a point on the line segment between   A   and   B. Then by Theorem 2.1 there is a real number   t   such that

C = (1 - t)A + tB

= ((1 - t)x1 + tx2, (1 - t)y1 + ty2)

and by the definition of a line segment,   0 < t < 1.

If we substitute these coordinates into the left side of the equation for the line, we get

a[(1 - t)x1 + tx2] + b[(1 - t)y1 + ty2]

= (1 - t)(ax1 + by1) + t(ax2 + by2)

But since   0 < t < 1,   both   t   and   1 - t   are positive so this is

< (1 - t)c + tc

= c

and   C   is on the same side of the line as   B.

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