Properties of the Circle. 397 



Connect the points D and F, by the right line DF ; K and G, 

 'by KG; and H and N, by HN. From the point L, whore the 

 diameter bisects DF, draw LM, parallel to EH or DK, or FG. 

 From M, the point where LM intersects KG, draw AIR, paral- 

 lel to HN. Through F, draw FP, parallel to EN; and through 

 P, the point where FP meets the circumference, draw PQ, 

 parallel to FD. 



Since DK and FG are parallel, and DL = LF, and LM is 

 is drawn parallel to DK and FG, it follows thatKM=:.MG; 



whence LM =_ — IH . It is manifest also, from the conrtruc- 



2 



lion, that the arch DEF is the common difference of the as- 

 sumed arches, and that the line FP, parallel to EN, cuts off the 

 arch FP equal to its supplement, since the angle DFP is a right 

 angle. It is obvious also, that PQ,, parallel to FD, meets the 

 diameter EN, in the same point K, in which MR, parallel to 

 HN, meets it. For, from the centre O, draw OT, perpendicu- 

 lar to LM. Since in the triangle LMR, OT is drawn parallel 

 to one of its sides, LT : TM 1: LO : OR. But LT = TM ; 

 therefore LO = OR ; of course, EL = NR. But EL = NQ ; 

 wherefore NR = NQ,. 



Then in the similar triangles LMR, EHN, it will be, EN : 



(LR or) FP : : EH : (LM or) 2-^i±Z5. Q. E. B. 



Corollary I. Hence the chord FG = — — — — — DK = 



EN 



2J^P.EH-DK;EN j^Hke manner, DK^^E^l^^^^^P-l^-^. 



EN LN 



Therefore, if the chord of the mean of three equidifferent arch- 

 es, be multiplied by twice the supplementary chord of their 

 common difference ; and from their product, there be taken the 

 product of the diameter and the chord of one of the extreme 

 arches; and the difference be divided by the diameter; the quo- 

 tient will express the chord of the other extreme arch. 



Cor. II. Hence the chord of any muUiple of an arch may 

 be expressed, in terms of the chord of the arch ; its supplemen- 

 tary chord ; and the diameter. 



Let the points D and K be supposed to coincide ; and then 

 let A denote the arch EDH. Then will the arch FDG = 2A. 



2FP-h^H 

 Whence the chord of 2A = —--=-, which is the expression 



EN 

 for the chord of the arch FDKG, when the arch DK = o. 



In like manner, considering A, 2A, 3A. as the arches, tlifi 



X 



