Theorem 2.14: (The second Pasch property) Let   A,   B,   and   C   be three noncolinear points. If a line crosses the line segment   AB,   then the line will either intersect line segment   AC,   segment   BC,   or go through point  C.

Proof: Let   D   be the point where the line crosses line segment   AB.   We can conclude from Theorem 2.1 that there are points on the line segment between   A   and   B,  which are on either side of the point   D.   By Theorem 2.5, then, there are points on that line segment which are on either sides of the line determined by   A   and   B.   Let   E   be a point on the same side of the line determined by   AB   as   C.

Consider the line determined by   CD.   Since   C   is, by hypotheses, not on the line determined by   AB,   we conclude that   CD   crosses   AB   at   D.   As a result, by Theorem 2.5,   A   is on one side of the line determined by   CD,   and   B   is on the other side.

If we substitute the coordinates of   E   into the equation for the line determined by   CD,  the trichotomy law for real numbers states that there are three cases.

Case 1:   E   is on   CD.   Then the line determined by   CD   and the line determined by   DE   both contain the distinct points   D   and   E,   so by Theorem 1.4, they are the same line, and the line determined by   DE   contains the point   C.

Case 2:   E   is on the same side of   CD   as   A.   We also have that   E   is on the same side of   AD   as   C,   because   E   is supposed to be on the same side of   AB   as   C,   and   AD   has the same equation as   AB,   so   E   is on the same side of   AD   as   C.   The hypotheses of Theorem 2.13 are satisfied, so we can conclude that   DE   crosses the line segment between   A   and   C.

Case 3:   E   is on the same side of   CD   as   B.   This case is the same as Case 2 if we simply switch   A's   and   B's,   and the conclusion is that   DE   crosses the line segment between   B  and   C.

3. Circles