000RM.dvi
24.3 Abundant and deficient numbers 631 Appendix: Mersenne primes Primes of the formMk=2k− 1 are called Mersenne prime. The only ...
632 Perfect numbers Appendix: Three important number theoretic functions Letnbe a positive integer with prime factorization n= ∏ ...
24.3 Abundant and deficient numbers 633 24.3.1 Appendix: Two enumerations of the rational numbers in (0,1) Consider two enumerat ...
634 Perfect numbers Exercise 1.The isle of Pythagora, while very sparsely populated, is capable of supporting a population of th ...
Chapter 25 Routh and Ceva theorems 1 Routh theorem: an example 2 Routh theorem 3 Ceva theorem and coordinates of triangle center ...
702 Routh and Ceva theorems 25.1 Routh theorem: an example ........... Given a triangleABC,X,Y,Zare points on the side lines spe ...
25.2 Routh theorem 703 25.2 Routh theorem ................. λ^1 μ 1 ν 1 A B X C Z Y P Q R We make use ofhomogeneous barycentric ...
704 Routh and Ceva theorems 25.3 Ceva Theorem ................. Theorem 25.1(Ceva).LetX,Y,Zbe points on the linesBC,CA,AB respec ...
25.3 Ceva Theorem 705 Example: incenter IfAX,BY,CZare the angle bisectors, the intersection is the incenter I: c b a c b a A B C ...
706 Routh and Ceva theorems Example: Gergonne point IfX,Y,Zare the points of tangency of the incircle with the sidelines, the li ...
25.3 Ceva Theorem 707 Example: Nagel point IfX,Y,Zare the points of tangency of the excircles with the respective sidelines, the ...
708 Routh and Ceva theorems Exercise 1.Calculate the area of trianglePQRgiven (a) λ=μ=ν=2. (b)λ=1,μ=7,ν=4; (c) λ=3,μ=7,ν=6. 2.Ca ...
25.3 Ceva Theorem 709 Appendix We give those values ofλ,μ,νwith numerators and denominators< 10 for which the area of triangl ...
710 Routh and Ceva theorems ...
Chapter 26 The excircles 1 Feuerbach theorem 2 A relation among radii 3 The circumcircle of the excentral triangle 4 The radical ...
712 The excircles 26.1 Feuerbach theorem .................. The nine-point circle is tangent internally to the incircle and exte ...
26.2 A relation among the radii 713 26.2 A relation among the radii ........... ra+rb+rc=4R+r. ra r rc rb D I A B C Ic Ia Ib O M ...
714 The excircles 26.3 The circumcircle of the excentral triangle .... The circle through the excenters has center at the reflec ...
26.4 The radical circle of the excircles 715 26.4 The radical circle of the excircles ........... The circle orthogonal to each ...
716 The excircles 26.5 Apollonius circle: the circular hull of the excircles I′ A B C Fc Fa Fb Fa′ fb′ Fc′ N ...
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