polar circle (d), nine point circle (t), circumcircle (e), circumcircle of the tangential triangle (s)
where A, B, C denote both the triangle's vertices and the angle measures at those vertices, H is the orthocenter (the intersection of the triangle's altitudes), D, E, F are the feet of the altitudes from vertices A, B, C respectively, R is the triangle's circumradius (the radius of its circumscribed circle), and a, b, c are the lengths of the triangle's sides opposite vertices A, B, C respectively.[1]: p. 176
The first parts of the radius formula reflect the fact that the orthocenter divides the altitudes into segment pairs of equal products. The trigonometric formula for the radius shows that the polar circle has a real existence only if the triangle is obtuse, so one of its angles is obtuse and hence has a negative cosine.
polar, circle, geometry, geometry, polar, circle, triangle, circle, whose, center, triangle, orthocenter, whose, squared, radius, ispolar, circle, triangle, polar, circle, nine, point, circle, circumcircle, circumcircle, tangential, triangle, displaystyle, beg. In geometry the polar circle of a triangle is the circle whose center is the triangle s orthocenter and whose squared radius isPolar circle red of a triangle ABC polar circle d nine point circle t circumcircle e circumcircle of the tangential triangle s r 2 H A H D H B H E H C H F 4 R 2 cos A cos B cos C 4 R 2 1 2 a 2 b 2 c 2 displaystyle begin aligned r 2 amp HA times HD HB times HE HC times HF amp 4R 2 cos A cos B cos C 4R 2 frac 1 2 a 2 b 2 c 2 end aligned where A B C denote both the triangle s vertices and the angle measures at those vertices H is the orthocenter the intersection of the triangle s altitudes D E F are the feet of the altitudes from vertices A B C respectively R is the triangle s circumradius the radius of its circumscribed circle and a b c are the lengths of the triangle s sides opposite vertices A B C respectively 1 p 176 The first parts of the radius formula reflect the fact that the orthocenter divides the altitudes into segment pairs of equal products The trigonometric formula for the radius shows that the polar circle has a real existence only if the triangle is obtuse so one of its angles is obtuse and hence has a negative cosine Properties EditAny two polar circles of two triangles in an orthocentric system are orthogonal 1 p 177 The polar circles of the triangles of a complete quadrilateral form a coaxal system 1 p 179 A triangle s circumcircle its nine point circle its polar circle and the circumcircle of its tangential triangle are coaxal 2 p 241 References Edit a b c Johnson Roger A Advanced Euclidean Geometry Dover Publications 2007 orig 1960 Altshiller Court Nathan College Geometry Dover Publications 2007 orig 1952 External links EditWeisstein Eric W Polar Circle MathWorld Retrieved from https en wikipedia org w index php title Polar circle geometry amp oldid 988241971, wikipedia, wiki, book, books, library,
