Theorem 3.8: If a line is tangent to a circle, then all of the points which are either on the circle or inside the circle except for the point of tangency are all on the same side of the line.

Proof: It will suffice to show that all of the points which are on the other side of the line from the center of the circle are outside the circle.

Let   O   be the center of the circle, let   P   be the point of tangency between the line and the circle, and let   A   be any point on the other side of the line from   O.   Since   O   is on one side of the tangent line and   A   is on the other side, there is some point where the line between   A   and   O   meets the tangent line, by Theorem 2.8. Call this point  B.

By Theorem 2.8,   B   is between   O   and   A.   So

|OA| = |OB| + |BA|

by Theorem 2.3, but by Theorem 3.7, the line which is tangent to the circle at   A   is perpendicular to the radius from   O   to   A,   and, by Theorem 3.6,

|OP| < |OB|

with equality if and only if   P = B.   Since   |BA|   is nonnegative we have

|OA| = |OB| + |BA| > |OB| > |OP| > r

so   A   is outside the circle.

next theorem (3.9)