This is one of Euclid's propositions which has been attributed to Thales of Miletus, who preceded Euclid by about 300 years. Euclid has a proof by contradiction. This sketch tries to show why the proposition is true rather than demonstrate Euclid's proof. David Joyce has a dynamic approach to Proposition 1.26 at [url]http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI26.html[/url], and Oliver Byrne has a beautiful, color, 19th century illustration of Euclid's proof at [url]http://www.math.ubc.ca/~cass/Euclid/book1/images/bookI-prop26.html[/url].
The proposition rather clunkily combines what we call Angle-Angle-Side congruency with Angle-Side-Angle.
