Easy Lemmas

Today I worked on exercises in Analysis with an Introduction to Proof and did a little mathematical puttering.

I’m also ready today to start sharing my work on the proposition about the inscribed equilateral triangle. I’ll start by proving that a triangle is equilateral if and only if its three angles are equal.

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First, recall from my post Revenge of the Squares, Part 1 that the angles opposite (i.e. subtended by) the equal sides of an isosceles triangle are also equal. The proof of the following related lemma is very similar.

Lemma: If two angles of a triangle are equal, then the sides that subtend them are also equal.

Consider the triangle $\triangle ABC$ with $\angle ACB\cong\angle ABC$.

Draw a line bisecting $\angle BAC$ and extend it to intersect $\overline{BC}$. Let the point of intersection be called $D$.

Now, by the angle-angle-side property, $\triangle ACD\cong\triangle ABD$, since $\angle ACB\cong\angle ABC$, $\angle CAD\cong\angle BAD$, and $\overline{AD}$ is shared.

Therefore, $AC=AB$. $\overline {AC}$ and $\overline {AB}$ are the sides that subtend the equal angles $\angle ACB$ and $\angle ABC$, so the lemma is proven.

(Note that I have added AAS to my list of triangle congruence properties to prove.¹)

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Proposition: A triangle is equilateral if and only if its three angles are equal.

Consider a triangle $\triangle ABC$.

Assume that $\triangle ABC$ is equilateral.

Then, since $AB=AC$, $\angle ACB\cong\angle ABC$, by the theorem cited above. Similarly, since $AC=BC$, $\angle ABC\cong\angle BAC$.

It follows that $\angle ACB\cong\angle ABC\cong\angle BAC$.

Thus, if $\triangle ABC$ is equilateral, then its three angles are equal.

Assume instead that the three angles of $\triangle ABC$ are equal.

Then, by the lemma, since $\angle ACB\cong\angle ABC$, $AB=AC$. Similarly, since $\angle ABC\cong\angle BAC$, $AC=BC$.

It follows that $AB=AC=BC$.

Thus, if the three angles of $\triangle ABC$ are equal, then it is equilateral.

This proves the equivalence.

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Stay tuned tomorrow for some circle geometry.

1. To do. ↩︎