342] 
THREE GIVEN POINTS AND TOUCH A GIVEN LINE. 
261 
Ihe triangle and the three triangles applied to the three sides form together a 
triangle similar to the original triangle but of double the linear magnitude, and the 
iorm of the curve of centres depends as has been shown on the position of the given 
line in regard to the triangle and the double triangle. The cases to be considered 
are tolerably numerous, but it is easy from the foregoing considerations, to see in any 
particular case what is the form of the curve of centres; for facility of delineation 
I select a form without infinite branches, see fig. 3, in which the given line cuts the 
two sides CA, CB, and the third side AB produced; it is moreover to be observed 
Fig. 3. 
C 
that as the figure is drawn the given line cuts the two sides CA, CB below their 
middle points Q and P respectively. By what precedes it appears that the middle 
points Q, P of these two sides CA, CB are each of them crunodes, but that the 
middle point R of the remaining side AB is an acnode. And this being so the 
general form of the curve is at once perceived to be that shown by fig. 3. 
It is very interesting to trace the corresponding positions of the point of contact 
on the given line, and of the centre on the curves of centres. When the point of 
contact is at oo, the centre is at I, as the point of contact moves from oc to q, the 
centre moves from I to q, and at q the two coincide ; as the point of contact moves 
from q to a point Q,, the centre moves from q to Q (along the branch Q2); as the 
point of contact moves from Q. 2 to a point P u the centre moves from Q to P (along 
the branch Q21P); as the point of contact moves from P 2 to p, the centre moves 
from P to p (along the branch PI) and at p the point of contact and the centre 
again coincide; as the point of contact moves from p to r, the centre moves from 
p to r and at r they again coincide; as the point of contact moves from r to a 
point P, the centre moves from r to P (along the branch 2P); as the point of contact 
moves from P, to a point Q 1} the centre moves from P to Q (along the branch 
P2 1Q) and finally as the point of contact moves from Q, to oc, the centre moves 
from Q (along the branch Q 2 ) to I, thus completing the circuit.