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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 $D*M*E$

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

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

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    $\begingroup$ Please change the title of your question - proposition 3.9a is not something people will know. $\endgroup$ – Piotr Benedysiuk Mar 15 at 17:18
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Then by the definition of line segment, either $M=C$, or $D*M*E$

This step is not correct/justified.

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.

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