A proof of euclids 47th proposition using the figure of the point within a circle and with the kind assistance of president james a. In this paper i offer some reflections on the thirtysecond proposition of book i of euclids elements, the assertion that the three interior angles of a triangle are. The same theory can be presented in many different forms. How to construct a line, from a given point and a given circle, that just touches the circle. The elements is a mathematical treatise consisting of books attributed to the ancient greek mathematician euclid in alexandria, ptolemaic egypt c. If a straight line touches a circle, and from the point of contact there is drawn across, in the circle, a straight line cutting the circle, then the angles which it makes with the tangent equal the angles in the alternate segments of the circle. In the first proposition, proposition 1, book i, euclid shows that, using only the. Guide now it is clear that the purpose of proposition 2 is to effect the construction in this proposition.
Proposition 32 in any triangle, if one of the sides is produced, then the exterior angle equals the sum of the two interior and opposite angles, and the sum of the three interior angles of the triangle equals two right angles. Built on proposition 2, which in turn is built on proposition 1. Begin by reading the statement of proposition 2, book iv, and the definition of segment of a circle given in book iii. Avigads work has been partially supported by nsf grant. Leon and theudius also wrote versions before euclid fl.
It is a collection of definitions, postulates, propositions theorems and. Axiomness isnt an intrinsic quality of a statement, so some. Let abcbe a triangle, and let one side of it bcbe produced to d. Remarks on euclids elements i,32 and the parallel postulate. If a straight line touches a circle, and from the point of contact there is drawn across, in the circle, a straight line cutting the circle, then the angles. Euclids elements, book iii, proposition 32 proposition 32 if a straight line touches a circle, and from the point of contact there is drawn across, in the circle, a straight line cutting the circle, then the angles which it makes with the tangent equal the angles in the alternate segments of the circle. The opposite segment contains the same angle as the angle between a line touching the circle, and the line defining the segment. The first three books of euclids elements of geometry from the text of dr. If two lines within a circle do no pass through the centre of a circle, then they do not bisect each other. The next two propositions depend on the fundamental theorems of parallel lines.