MONT 104Q – Mathematical Journeys: From Known To Unknown Working Within Euclid’s System – September 30, 2015

Background

By this time we have covered up to Proposition 16 in class, and you have read through Proposition 22 in Book I of the Elements. To recap, recall that we know the Common Notions and Postulates, a number of basic constructions, as well as the following facts:

Proposition 12: Given a line and a point not on the line we can construct a perpendicular from the point to the line. Proposition 15: Vertical angles at the intersection of two straight lines are equal Proposition 16: An exterior angle of a triangle is greater than either nonadjacent interior angle Proposition 17: The sum of each pair of interior angles in a triangle is less than two right angles (i.e. less than 180^{∘} for us). Proposition 18: If in a triangle one side is greater than another side, then the angle opposite the greater side is greater than the angle opposite the other side. Proposition 19: If in a triangle one interior angle is greater than another interior angle, then the side opposite the greater angle is greater than the side opposite the other angle. Proposition 20: (the “triangle inequality”) The sum of any two sides of a triangle is greater than the other side.

Even though Euclid never quite states this, he uses and we will also use an additional “common notion” today:

(“CN 6”) If one thing is greater than a second and equals are added to both, then the first sum is greater than the second sum.

Side trips on our main journey

Using only (I repeat, only!) what we know at this point, prove the following, with justifications for every step.

A) (A “warm-up”) Using Proposition 16, prove Proposition 17. (Hint: Let the triangle be ΔABC. “Without loss of generality” we can consider the two interior angles at A and B. Extend the side CB to CD and apply Proposition 16 to the exterior angle < DBA.)

B) Let ΔABC be an equilateral triangle and let D be a point on the side BC different from B, C. Prove that the line segment AD is shorter than the sides of the triangle.

C) Prove the following important fact: Given a line and a point P not on the line, the perpendicular from P to the line (the result of the construction from Proposition 12) is the shortest line segment joining P to any point on the line. (Note: You will probably be tempted to use the Pythagorean Theorem here. We have not proved that!)