ON SOME PROPERTIES OF THE PARABOLA. 65 



(1) o 8 -(2) 4 = 0, 

 and (1) 3 - (2) cq = 0, 



and which have a point in common, cut (3) in points whose ab- 

 scissas are equal, and which therefore coincide. Hence either 

 two straight lines enclose a space, or (3) passes through the in- 

 tersection of (1) and (2). Thus the existence of the point 

 common to the three diagonals has been proved, and its abscissa 

 found. To determine its ordinate, add (1), (2), (3), when x dis- 

 appears, and we have 



y = m foa, (a, + a 5 ) - <? 2 3 (a, + a 6 ) + 3 a 4 (or e + aj - a 4 a s (a, + Cf 8 ) 



+ a 5 * 6 (a 2 + a3 )+a 6 a 1 ( a3 + a 4 )} 

 divided by 



a i?2 - 2 a 3 + V4 - a 4 5 + VG ~ *1' 



If we call the co-ordinates of the point where the third and 

 sixth sides of the hexagon meet o\ u y su , and so of the other two 

 points, these expressions for x and y become 





These expressions, as of course we should expect, are sym- 

 metrical. 



In the last Number of this Journal a demonstration was given 

 of a property of a parabola : That the circle which passes through 

 the intersections of three tangents also passes through the focus. 

 Although six demonstrations of this theorem have already ap- 

 peared, yet the following is so simple that its insertion here may 

 not be inappropriate. 



Referring the parabola to the focus as origin, we can put the 

 equation to the tangent under the form 



_^= f m -\ 

 ** m \ in) ' 



where a is one-fourth of the parameter, and m the trigonometri- 

 cal tangent of the angle which the tangent makes with the axis 

 of y . Hence if x t , y l be the co-ordinates of the point of inter- 

 section of 



x f l\ 



= a (m + ) , 

 m \ m) 



y -- = a 

 y 



