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Euler Inequality
In geometry, Euler's theorem states that the distance ''d'' between the circumcenter and incenter of a triangle is given by d^2=R (R-2r) or equivalently \frac + \frac = \frac, where R and r denote the circumradius and inradius respectively (the radii of the circumscribed circle and inscribed circle respectively). The theorem is named for Leonhard Euler, who published it in 1765. However, the same result was published earlier by William Chapple in 1746. From the theorem follows the Euler inequality: R \ge 2r, which holds with equality only in the equilateral case. Stronger version of the inequality A stronger version is \frac \geq \frac \geq \frac+\frac+\frac-1 \geq \frac \left(\frac+\frac+\frac \right) \geq 2, where a, b, and c are the side lengths of the triangle. Euler's theorem for the escribed circle If r_a and d_a denote respectively the radius of the escribed circle opposite to the vertex A and the distance between its center and the center of the circumscribed circl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Theorem2
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus. He introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory. Euler is held to be one of the greatest mathematicians in history and the greatest of the 18th century. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." Carl Friedrich Gauss remarked: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it." Euler is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Escribed Circle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides. The center of the incircle, called the incenter, can be found as the intersection of the three internal angle bisectors. The center of an excircle is the intersection of the internal bisector of one angle (at vertex , for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex , or the excenter of . Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
