328 A. Mukhopadhyay — Memoir on Plane Analytic Geometry. [No. 3, 

 (QE-PF)2-4c[q. sin (^-aWp | cos (^-a)-cos aHe 



-4c [P sin f^- a) - Q I cos ^^- aWcos a| ] F + 4c2 sin^ A = 0. 



As E and F are linear functions of x and y, while P and Q are constant 

 quantities, it is clear that this equation of the required envelope 

 represents a parabola, and a diameter of this parabolic envelope is given 

 by QE = PP, 



■which is equivalent to 



jP.cos ^y-A^ + Qsin (^-^\x- | P cos^- Q sin ^| ?/ = 0, 



or, since 





r2. A . . /2. 



and 



P cos - - Q sin - = 



. -(i+^) 



3 ^ 3 sinB ' 



the equation of the diameter may be written 



X sin ^^+ C^-y sin (^+b) = 0. 



The diameter can be geometrically constructed as follows, viz., on BO 

 describe externally an equilateral triangle BDO, and join AD ; then AD 

 is the diameter ; for, if the point D be {x, y), we have 

 DC si n A DB _ sin A 



^^i.(3+oy -^ing^.B)• 



so that the equation of AD is 



» sin ^^+ Cj = 2/ sill (i+^)' 

 which is also the equation of the diameter. 



Again, if we consider the envelopes of the other two sides, they also 

 will be parabolas, and their diameters will be obtained by joining B and 

 C to the remote vertices E and F of the equilateral triangles described 

 externally on the opposite sides ; and, since, from elementary geometry, 

 AD, BE, OF intersect in a point, it can easily be shewn, from Euc. III. 22, 



that the acute angle between any two of them is -. Thus, finally, we 



o 



have the 



Theorems. — The envelopes of the sides of the equilateral triangles 



which can be inscribed in any given triangle ABC, are three parabolas ; 



