...same number of similar triangles, having the same ratio to one another that the polygons have. 11. The rectangle contained by the diagonals of a quadrilateral inscribed in a circle, is equal to both the rectangles contained by its opposite sides. No. 6. 1. If from the ends of a side of a triangle,...

...but not as clearly understood. Let OA,,OAirOAt, &c.=a,,a»,°s» &c. Now by Ptolemy's Theorem : — The rectangle contained by the diagonals of a quadrilateral inscribed in a circle &c., we easily get the following relations: c(«4+ae ) t )—dai c(a, -fa,, ) c(at Hence d-{ (at+at+at+d1+at+ai^+at^—...

...sides. 10. The rectangle contained by the diagonals of a quadrilateral 11 inscribed in a circle ia equal to the sum of the two rectangles contained by its opposite sides. An arc AB of a circle is bisected at С and P is any point on the arc ; eliew that the difference between...

...the subject he givei there the theorem afterwards inserted in Euclid (Book VI., Prop. D) relating to the rectangle contained by the diagonals of a quadrilateral inscribed in a circle. The Arabians made the improvement of using in place of the chord of an arc the sine, or half chord...

...KuVAu tvticiav). The theorem may be enunciated thus. The rectangle contained by the diagonals of any quadrilateral inscribed in a circle is equal to the sum of the rectangles contained by the pairs of opposite sides. I shall give the proof in Ptolemy's words, with...