1824.] connected with the Trisection of an Arc. 363 



Dem. — Draw the radii A c, A k. The triangle c k D is similar 

 to the triangle c D E, each being similar to dnD. .-. E c, c D, 

 c k are in continued proportion, that is, E c, c A, c &. /. the 

 triangle c A; A is similar to c A E. .•. angle c A /c = A E c = 



— — . But angle A c b, or A B o, = c A k + c k A = 3 c A & 



= 3 A E c. .-. ano;le A B a = — — A 



Prop. VIII. (Fig. 10.) 



If the angles A E a, A e a be two-thirds respectively of the 

 angles A B a, Aba, and if there be drawn from E and e the 

 lines E T, e t, each making an angle of 30° with the axis ; these 

 lines meet each other in the periphery of the circle A B a b. 



Dem. — The lines E T, e t trisect the arcs A B a i Aba 

 respectively in T and t (see Cor. 1, Prop. III). Let the line ET 

 meet the circle again in C, and draw the ordinate C c. The 



angle C B c = 60° + ^- [for C B b = C E b + BCE = 30° 

 + — g— .*. &c.J. In like manner, the angle at b subtended by 

 the ordinate drawn through the point in which e t again meets 



the circle is equal to 60° -\ — . But the two angles 60° + 



— — - and 60° -\ — are supplemental to each other, the angles 



—^- and — — - being together equal to 60°. Therefore, the for- 

 mer angle [60° H ^-\ being equal to C B c, the latter (60° + 



■■ ■ ■-) must be equal to C b c. /.the ordinate drawn through the 



point in which e t again meets the circle coincides with C c, and 

 e t meets E T in C. 



Cor. 1. — The sum of the angles A E e, A e E being 60° [as 

 they are respectively the thirds of two supplemental angles], it 

 follows that the angle E A e = 120° = E C e. Therefore a 

 circle described with the centre c and radius c C must circum- 

 scribe the triangle EAe, And in that circle E e is the side of 

 an equilateral triangle. 



Cor. 2. — Producing A a to meet the circle A E D e in d, the 

 angle A E d = 30° + A E a. For angle c E A = D E d, each 

 of them being equal to C E A. .*. angle dE« = DEca 30°. 

 But angle A E d = d E a + A E a = 30 9 + A E a. 



From the principles which have been established, we should 

 be warranted immediately to infer the following proposition. 

 But I think it expedient to demonstrate its truth independently 

 of the preceding theorems. 



