THE CLASSICAL THEOREMS OF MENALAUS, CEVAS AND DESARQUES

Authors

  • Susan C. Vallejo

Abstract

The study showed the proofs of the classical theorems of Menalaus, Cevas and Desarques using
the Fixed Point Theorems in Euclidean Geometry. Properties of transformations, translations and central
dilation were applied to prove several fixed point theorems.
This paper is expository in nature, concepts were precisely defined and illustration were given to
concretize the concepts for a better comprehension of the theory involved. Proofs to lemma, corollaries and
theorems were presented as complete as possible and made accessible to the reader by diagrams.
Based from the proofs in this study, it was found out that for any two points x, y on a
plane. D(x) D(y) are the images of x, y on a plane, D(x) and D(y) are the image of x, y under the dilation D.
Moreover, D1D2 are dilations such that D1(x) = D2(x) and D1(y) = D2(y) then D1 = D2 . For a dilation D with
at least two fixed points, D is the identity mapping. Moreover, it was proven that a translation is a dilation.
Topics in Euclidean Geometry dealing with transformation can be employed to bring theorems from
classical synthetic geometry into the so-called mainstream of modern mathematics
Transformation proof of more familiar theorems of geometry like; pappus theorem, the nine-point circle
theorem, Euler theorem may be considered as another study

Published

2020-04-30

Issue

Section

Articles