Exterior angle theorem | EPFL Graph Search

The exterior angle theorem is Proposition 1.16 in Euclid's Elements, which states that the measure of an exterior angle of a triangle is greater than either of the measures of the remote interior angles. This is a fundamental result in absolute geometry because its proof does not depend upon the parallel postulate. In several high school treatments of geometry, the term "exterior angle theorem" has been applied to a different result, namely the portion of Proposition 1.32 which states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. This result, which depends upon Euclid's parallel postulate will be referred to as the "High school exterior angle theorem" (HSEAT) to distinguish it from Euclid's exterior angle theorem. Some authors refer to the "High school exterior angle theorem" as the strong form of the exterior angle theorem and "Euclid's exterior angle theorem" as the weak form. A triangle has three corners, called vertices. The sides of a triangle (line segments) that come together at a vertex form two angles (four angles if you consider the sides of the triangle to be lines instead of line segments). Only one of these angles contains the third side of the triangle in its interior, and this angle is called an interior angle of the triangle. In the picture below, the angles ∠ABC, ∠BCA and ∠CAB are the three interior angles of the triangle. An exterior angle is formed by extending one of the sides of the triangle; the angle between the extended side and the other side is the exterior angle. In the picture, angle ∠ACD is an exterior angle. The proof of Proposition 1.16 given by Euclid is often cited as one place where Euclid gives a flawed proof. Euclid proves the exterior angle theorem by: construct the midpoint E of segment AC, draw the ray BE, construct the point F on ray BE so that E is (also) the midpoint of B and F, draw the segment FC. By congruent triangles we can conclude that ∠ BAC = ∠ ECF and ∠ ECF is smaller than ∠ ECD, ∠ ECD = ∠ ACD therefore ∠ BAC is smaller than ∠ ACD and the same can be done for the angle ∠ CBA by bisecting BC.

On Random Tomography with Unobservable Projection Angles

Victor Panaretos

We formulate and investigate a statistical inverse problem of a random tomographic nature, where a probability density function on

\mathbb{R}^3

is to be recovered from observation of finitely many of its two-dimensional projections in random and unobserv ...

Institute of Mathematical Statistics2009