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Why is that if a triangle is not isosceles, the external bisectors cross the opposite (extended) sides of the triangle?

wondering1123
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1 Answers1

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Let's have the non-isosceles triangle $ABC$ and we have the external bisector at $A$ being parallel to $BC$. Then you form two equal angles $\alpha$ at $A$. Since the bisector is parallel to $BC$, $\angle ABC=\alpha$. But $2\alpha$ is an external angle, so $$2\alpha=\angle ABC+\angle ACB=\alpha+\angle ACB$$ This means $$\angle ACB=\alpha=\angle ABC$$ so the triangle is isosceles, which contradicts the hypothesis.

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Andrei
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