Steiner-Lehmus Theorem/Lemma 1

Lemma for Steiner-Lehmus Theorem

In the same circle, let one chord $PR$, be larger than another, $PQ$.

Then the inscribed angle subtended by $PR$, with its vertex in the major arc, is larger.

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Proof

Let there be a circle on center $O$.

Let $PQ$ and $PR$ be two chords on the circle from the same point $P$.

Let chord $PR > PQ$.

Let arc $PQ$ be the minor arc of the chord $PQ$.

Without loss of generality, let arc $PR$ be the arc that contains arc $PQ$.

By Common Notion $5$:

arc $PR > $ arc $PQ$

Hence the central angles are such that:

$\angle POR > \angle POQ$

In the words of Euclid:

In a circle the angle at the centre is double of the angle at the circumference, when the angles have the same circumference as base.

(The Elements: Book $\text{III}$: Proposition $20$)

Then an inscribed angle subtended by $PR$ is greater than an inscribed angle subtended by $PQ$

The result follows.

$\blacksquare$