# If a ray r emanating from an exterior point of triangle ABC intersects side AB at any point between A and B, then r also intersects side AC or side BC [duplicate]

Prove Proposition 3.9: If a ray r emanating from an exterior point of triangle $$ABC$$ intersects side $$AB$$ at any point between $$A$$ and $$B$$, then $$r$$ also intersects side $$AC$$ or side $$BC$$.

Can someone help me finish this proof I started

By the hypothesis, let $$DE$$ be a ray emanating from an exterior point $$D$$ of triangle $$ABC$$ where the ray $$DE$$ intersects $$AB$$ at point $$E$$ such that $$A*E*B$$.

By pachs theorem, the line DE either intersects $$AC$$ or $$BC$$. 3.Suppose $$DE$$ intersects $$AC$$ at point $$M$$ where $$M$$ is not equal to $$A$$ Then by the definition of line segment, either $$M=C$$, or $$D*M*E$$

## marked as duplicate by Lee David Chung Lin, Leucippus, Lord Shark the Unknown, Vinyl_cape_jawa, uniquesolutionMar 16 at 9:57

This question was marked as an exact duplicate of an existing question.

• Please change the title of your question - proposition 3.9a is not something people will know. – Piotr Benedysiuk Mar 15 at 17:18

Then by the definition of line segment, either $$M=C$$, or $$D*M*E$$
Up to that point your proof is correct. In fact, the only thing you need to conclude the proof is that $$D*E*M$$ or $$D*M*E$$. Since these points are distinct, you may assume the contrary i.e. $$E*D*M$$, and show that $$D$$ is an interior point of a triangle which contradicts the assumption. To do this, prove that $$D$$ belongs to all three halfplanes which determine the interior of a triangle.