In geometry , points that are on the same circle are referred to as cyclic (or homocyclic ) points. Three points on a plane that do not lie on the same line always lie on the same circle, therefore sometimes the term “cyclic” is applied only to sets of 4 or more points. [one]
Mid perpendiculars
In the general case, the center O of the circle on which the points P and Q lie must be such that the distances OP and OQ are equal. Therefore, the point O must lie on the median perpendicular (or on the mediatrix) of the segment PQ . [2] . A necessary and sufficient condition for n different points to lie on the same circle is that n ( n - 1) / 2 mediatrices of segments having any pairs of n points at their ends all intersect at the same point, namely: in the center O.
Inscribed Polygons
Triangles
The vertices of each triangle lie on a circle [3] . The circle passing through the 3 vertices of a triangle is called the circumscribed circle of the triangle. Several other sets of points, which are determined from a triangle, also lie on the same circle, that is, they are cyclic points; see Euler's Circle [4] and Leicester's Circle . [five]
The radius of the circle on which there are many points, by definition, is the radius of the circumscribed circle of any triangle with vertices at any three of these points. If the pairwise distances between any three of these points a , b, and c , then the radius of the circle is
The equation of the circumscribed circle for the triangle, and the expression for the radius and coordinates of the center of the circle through the Cartesian coordinates of the vertices are given here .
Quadrangles
A quadrangle ABCD with vertices lying on the same circle is called inscribed ; this happens if and only if (by the inscribed angle of the circle theorem), which is true if and only if the opposite angles of the quadrangle complement each other up to 180 degrees. [6] The inscribed quadrilateral with consecutive sides a , b , c , d and the semiperimeter s = ( a + b + c + d ) / 2 has a radius of the circumscribed circle equal to [7] [8]
This expression was obtained by the Indian mathematician in the 15th century.
By Ptolemy’s theorem , a quadrilateral defined by pairwise distances between its four vertices A , B , C and D, respectively, will be inscribed if and only if the product of its diagonals is equal to the sum of the products of opposite sides:
If two straight lines, one of which contains the segment AC , and the other contains the segment BD , intersect at one point "X", then these four points A , B , C , D are cyclic points if and only if [9]
The intersection point X can be both inside and outside the described circle. This theorem is known as the point power theorem.
n-gons
In the general case, an n -gon with all its vertices lying on one circle is called an inscribed polygon . A polygon is an inscribed polygon if and only if all the middle perpendiculars of its sides intersect at one point. [ten]
Notes
- ↑ Efremov, 1902 , p. 34.
- ↑ Libeskind, Shlomo (2008), Euclidean and Transformational Geometry: A Deductive Inquiry , Jones & Bartlett Learning, p. 21, ISBN 9780763743666 , < https://books.google.com/books?id=6YUUeO-RjU0C&pg=PA21 > /
- ↑ Elliott, John (1902), Elementary Geometry , Swan Sonnenschein & co., P. 126 , < https://books.google.com/books?id=9psBAAAAYAAJ&pg=PA126 > .
- ↑ Isaacs, I. Martin (2009), Geometry for College Students , vol. 8, Pure and Applied Undergraduate Texts, American Mathematical Society, p. 63, ISBN 9780821847947 , < https://books.google.com/books?id=0ahK8UneO3kC&pg=PA63 > .
- ↑ Yiu, Paul (2010), " The circles of Lester, Evans, Parry, and their generalizations ", Forum Geometricorum T. 10: 175–209 , < http://forumgeom.fau.edu/FG2010volume10/FG201020.pdf > .
- ↑ Pedoe, Dan (1997), Circles: A Mathematical View (2nd ed.), MAA Spectrum, Cambridge University Press, p. xxii, ISBN 9780883855188 , < https://books.google.com/books?id=rlbQTxbutA4C&pg=PR22 > .
- ↑ Alsina, Claudi & Nelsen, Roger B. (2007), " On the diagonals of a cyclic quadrilateral ", Forum Geometricorum T. 7: 147–9 , < http://forumgeom.fau.edu/FG2007volume7/FG200720.pdf >
- ↑ Hoehn, Larry (March 2000), “Circumradius of a cyclic quadrilateral,” Mathematical Gazette T. 84 (499): 69–70
- ↑ Bradley, Christopher J. (2007), The Algebra of Geometry: Cartesian, Areal and Projective Co-Ordinates , Highperception, p. 179, ISBN 1906338000 , OCLC 213434422
- ↑ Byer, Owen; Lazebnik, Felix & Smeltzer, Deirdre L. (2010), Methods for Euclidean Geometry , Mathematical Association of America, p. 77, ISBN 9780883857632 , < https://books.google.com/books?id=W4acIu4qZvoC&pg=PA77 > .
Literature
- Efremov Dm. The new geometry of the triangle. - Odessa, 1902.