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English: Animated visual proof of Ptolemy's theorem, based on W. Derrick, J. Herstein (2012) Proof Without Words: Ptolemy's Theorem, The College Mathematics Journal, v 43, n 5, p 386 Date 22 May 2022
Ptolemy's Theorem yields as a corollary a pretty theorem [2] regarding an equilateral triangle inscribed in a circle. Given An equilateral triangle inscribed on a circle and a point on the circle. The distance from the point to the most distant vertex of the triangle is the sum of the distances from the point to the two nearer vertices.
For four points in order around a circle, Ptolemy's inequality becomes an equality, known as Ptolemy's theorem: ¯ ¯ + ¯ ¯ = ¯ ¯. In the inversion-based proof of Ptolemy's inequality, transforming four co-circular points by an inversion centered at one of them causes the other three to become collinear, so the triangle equality for these three points (from which Ptolemy's inequality may ...
The Elements begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language. [1]
For any simple quadrilateral with given edge lengths, there is a cyclic quadrilateral with the same edge lengths. [ 43 ] The four smaller triangles formed by the diagonals and sides of a convex quadrilateral have the property that the product of the areas of two opposite triangles equals the product of the areas of the other two triangles.
Casey's theorem and its converse can be used to prove a variety of statements in Euclidean geometry. For example, the shortest known proof [ 1 ] : 411 of Feuerbach's theorem uses the converse theorem.
Proclus (410–485) wrote a commentary on The Elements where he comments on attempted proofs to deduce the fifth postulate from the other four; in particular, he notes that Ptolemy had produced a false 'proof'. Proclus then goes on to give a false proof of his own. However, he did give a postulate which is equivalent to the fifth postulate.
If we draw both circles, two new points are created at their intersections. Drawing lines between the two original points and one of these new points completes the construction of an equilateral triangle. Therefore, in any geometric problem we have an initial set of symbols (points and lines), an algorithm, and some results.