Journal of Mathematical Sciences: Advances and ApplicationsAuthor's: David Fraivert
Pages: [1] - [34]
Received Date: May 8, 2016
Submitted by:
DOI: http://dx.doi.org/10.18642/jmsaa_7100121666
Euclidean geometry is one of the oldest branches of mathematics –
the properties of different shapes have been investigated for
thousands of years. For this reason, many tend to believe that today
it is almost impossible to discover new properties and new directions
for research in Euclidean geometry.
In the present paper, we define the concepts of “Pascal points”,
“a circle that forms Pascal points”, and “a circle coordinated
with the Pascal points formed by it”, and we shall prove nine
theorems that describe the properties of “Pascal points” on the
sides of a convex quadrilateral.
These properties concern the following subjects:
● The ratios of the distances between the Pascal points formed on a
pair of opposite sides by different circles.
● The ratios of the distances between the centers of the circles
“that form Pascal points on the sides of the quadrilateral”, and
the ratios of the distances between the Pascal points formed using
these circles.
● Special types of circles “that form Pascal points on the sides
of a quadrilateral”.
● The properties of Pascal points and the centers of the special
circles defined.
using the general Pascal’s theorem, complex numbers methods in plane geometry, the theory of a convex quadrilateral and a circle that forms Pascal points, Pascal points, a circle that forms Pascal points, a circle coordinated with the Pascal points formed by it, harmonic quadruplet, inversion transformation, pole and its polar relative to a circle, Thales’ theorem, circle of Appolonius.
