362 Mr. Walker on some Geometrical Principles [MAy, 



fill on the hypotenuse ; and if with c as the centre, and the side 

 c T as radius, a circle be described, and another circle, with k as 

 the centre and the remote segment k E as radius, intersecting 

 the former in the points A and a ; the angles A E a and A B a 

 are as two to three. That the circles may intersect, the side cT 

 must be more than half the hypotenuse ; or the angle c E T must 

 be more than 30°. 



Prop. VI. (Fig. 8.) 



The same things being supposed, the line Y x, joining the 

 point of trisection in the arc Aba with the point in which the 

 other trisecting line meets the circle A E a e, is a tangent to 



the circle A Eae, and is equal to Y y, the chord of arc — — . 



For the line A Y is a tangent at A, the angle a A Y being equal 



to — — — = A E a. .*.A Y, or Y y, is a mean proportional 



between E Y and X Y, that is between E y and x y. .'. the tri- 

 angle y Y x is similar to the triangle Y E y. .'. Y \r = Y y, and 

 is a tangent at x, the angle ;iE being equal to E X x. 



Cor. ] . — Let there be drawn B m, B n parallel to the trisecting 

 lines ; and let & q be the base of an inscribed equilateral triangle 

 parallel to A a : — then the arc m a is the third part of z a r, and 

 therefore m q the third part of z A r. For the triangles YEy, 

 yYx being similar isosceles triangles, the angle EYi (stand- 

 ing on the arc z a r) = E Y y — Y E y = Bra » — mBr. 

 .*. arc x«r = nflB- 2 b n. But arc n a B = semicircular 



arc B b — b n. ,\ arc zar = Bb— 3bn. ;. '—=- = -a bn 



= b q — b n = n q t or m e. 



Cor. 2. — B m is equal to y r ; and y r passes through k, the 

 centre of the circle AE«e. For arc BAm=BAY + Y in 

 = ~Bay+~Br = r ay. .'. B»« = ry. But we have seen that 

 X y = ay. ,\ ky bisects aX. perpendicularly. /. angle a ky = 



" Q - = a E X. And angle a k e = a E A. .•. angle y k e = 



A E X. But the same is the value of the angle which ry makes 

 with the axis. For the sum of that angle and Y E b = Y r y 

 a AEJ = AEY + YEJ. ■'• r y passes through k. 



Prop. VII. (Fig. 9.) 



In the produced diameter of a given circle A B a b, let E be 

 any point whose distance from the vertex is less thaii radius ; 

 and from E to the extremities of the normal diameter, let there 

 be drawn the lines ED, Erf, cutting the circle in the points 

 m and n. Let k be the point in which the lines Tin, dm cut 

 the axis ; and with the centre k and the interval k E let a circle 

 be described cutting the given circle in the points A, a. Then 

 the angle A E a is two-thirds of the angle A B a. 



