Lecture 17 We finished talking about Book III, which deals with the properties of circles. We proved the following proposition which we stated last time: and we then commented the following proposition, which we proved long ago when talking about Thales (=a triangle inscribed in a semicircle is a right triangle): We then passed to (the much shorther) Book IV (7 Definitions and 16 Propositions). In this book Euclid describes the constructions for inscribed and circumscribed figures (in a circle). We proved the following propositions which tell us how to inscribe and circumscribe a given triangle (we spoke about two important points in a triangle: the incenter and the circumcenter): Proposition 4 please read the commentary about Heron's formula Proposition 5 please read the commentary about the law of sines. We then discussed the issue of inscribing a regular pentagon in a given circle. This is the most difficult construction done in Book IV. (The pentagon is related to the pentagram of the Pythagoreans). Euclids solves this problem by reducing it to the construction of an isosceles triangle having each of the angles at the base double the remaining one. (I.e. he shows how to construct a 36-72-72 degrees triangle). Please read the book by Professor Hartshorne (Section 4 of Chapter 1, pages 45-51) to see how to construct a regular pentagon in as few steps as possible: Finally, we discussed how to construct a regular hexagon and a regular 15-gon. We will make more comments about these issues next time: Click on the above links to go directly to the proofs and commentaries of the results.