78 The Three Sections, Tanyencies, 



scrijitable in a circle pass through fixed points A., B, C, D, E, 

 Y, G, and that the rectangles Aa.Ab^ Bb.Bc, S^^c, under the 

 distances of the fixed points from the angular points on the sides 

 through them are constants ; to find the relations amongst the 

 data so that the locus of a, any angle of the polygon, may he 

 found, ^'C. 



ANALYSIS. 



Let O be the centre of tlie circle ahcdefg. Then, since 

 AO- -racl^O and GO^ - rad- O are eqnals to M.kb and 

 Qg.Ga, .-. GO'^ — AO^ must be constant, and tlie perpendicu- 

 lar from O on AG must cut it in a fixed point. Similarly, 

 the perpendicular from O on BG must cut it also in a fixed 

 point. And hence we see that when any three A, G, B, of 

 the points are not in a straight line, the centre O of the circle 

 is fixed, as also the circle O itself. And in this case the locus 

 of a must be — if anywhere — in the circumference of the circle 

 O. 



But if the fixed points A, B, C, &c., be in a straight line, 

 it is evident the centre O is not a fixed point, and that its 

 locus is restricted to a straight line perpendicular to the line 

 ABC, &c., and that its circumference must pass through two 

 fixed points, real or imaginary, in the line ABC, &c. i\Iore- 

 over; it is evident that if we putX for the point in which the 

 perpendicular from O cuts AB, &c., and that Ave put x' , a', b' , 

 c, &c., for the distances of X, A, B, C, &c., from an assumed 

 point on ABC, &c., then must the following relations exist, 

 viz.: {a'-x'Y - {b'-x'Y = l-m, [a - xj - {c'-xy=l-n, 

 {a'-x'Y - {d'~x'Y = l-p, [a'-x'Y - {e'-x'f=l~q, («'- 

 x'Y - {f -x'Y=l-r, [a'-x'Y - {9'-x'Y=l-s; in which 

 I, m, n, p, q, r, s, represent the magnitudes of the constant 

 rectangles. And determining on any N + 2 of the 2ISr + 1 

 quantities represented we can easily find the remaining N — 1 . 

 Now looking on the three consecutive sides ab, be, cd, of the 

 polygon, we know (from involution property of inscriptable 

 quadrilateral) that when the points A, B, C, D, E, E, G are 

 in a straight line, ad passes through a fixed point in this line; 

 and looking on the three consecutive sides ad, de, and ef, we 

 know that af passes through a fixed point in the same line 

 ABC, &c. And in this manner we may evidently proceed 

 until we come to the last so formed quadrilateral having its 

 fourth side coincident with a side of the polygon, or infinitely 

 small, and tangent at a, just according as the number of sides 



