110 Mr WALLACE, ON GEOMETRICAL THEOREMS, AND FORMULAE, 



CAd and the common angle DAcl, therefore the triangles BAil, DAC 

 are similar. 



CoroUary. Let ABC be any triangle, and let straight lines AD. 

 BD, CD be drawn to D, any point in its plane; at the point B in 

 the line BA make the angle ABd equal to the angle ADC, viz. that 

 which the side opposite to B subtends at Z); at the point C in the 

 line CA make the angle ACd equal to the angle ADB, which AB the 

 side opposite to C subtends at D; draw a line from the remaining 

 angle A to d the intersection of the lines Bd, Cd; the triangles ADC, 

 ABd are similar; and the triangles ADB, ACd are similar. 



5. The truth of the Corollary may be inferred from the theorem: 

 it may however be proved directly as follows. 



Let E be the intersection of the lines dB, CD (fig. 1.); join AE. 

 Because by construction the angles ADC, ABd are equal, the angles 

 ADE, ABE are equal ; therefore the points A, D, B, E are in the 

 circumference of a circle; hence the angle AEB is either equal to the 

 angle ADB, or is its supplement ; now by hypothesis the angle ADB 

 is equal to ACd; therefore AEB or AEd is either equal to ACd, or 

 is its supplement ; hence, in each case, the points A, d, C, E are in the 

 circumference of a circle; and therefore the angle ACD or ACE is 

 equal to AdB; now, by construction, the angle ADC is equal to the 

 angle ABd; therefore the triangles ADC, ABd are similar; and since 

 the angle CAD is equal to dAB; by adding or subtracting the angle 

 dAD, we have the angle CAd equal to DAB; now the angle ACd is 

 by construction equal to ADB; therefore the triangles ACd, ADB 

 are similar. 



6. In the demonstration, it was assumed that when the angles 

 BAD, CAd are equal, then BAd, CAD are equal; this will always 

 be true when the lines AD, Ad are either both within the angle BAC, 

 or both without that angle, or, which is the same thing, when the 

 similar triangles BAD, dAC are similarly situated; and the same is 

 true of the similar triangles BAd, DAC. 



7. I shall now apply the geometrical theorem to the construction 

 of the problem enunciated in the beginning of this paper. 



