An elementary treatise on cubic and quartic curves, by A. B. Basset.

44

TANGENTIAL COORDINATES.

(a, /,, ) and (I, qr, ) satisfy the equations Xa + G/3 + vry =0 and
aa + b3 + cy = 2A, it follows that
x ( - ) + u (/3 - r) + v (y - )= 0,
a(a - ~)+ b (/3 - )+ c (y7- ) =0,
whence
ac-J - 7-
c - vb va-X Xb - a '
accordingly by (11) of ~ 22 the condition is
dF dF dF
(Yc - vb) + (va - Xc) + (Xb - /a) = 0.
72. The foregoing system of tangential coordinates may be
exhibited in a geometrical form. Let the line (X, pu, v) cut the
sides BC, CA, AB of the triangle of reference in D, E and F; and
let p, q, r be the lengths of the perpendiculars from A, B, and C
on to it; also let any two of these perpendiculars, say p and q, be
considered to have contrary signs when the line cuts AB at a
point lying between A and B, and in other cases to have the same
sign. Then if F lies between A and B,
q BF
p AF'
Putting 7y= 0 in the equation of DEF, we obtain
L _ a BFsinB qb
X i/- AFsinA pa'
whence X............... (29),
pa qb rc
which shows that X, ut, v are proportional to the products of the
lengths of each perpendicular into the lengths of the opposite
sides. The equation of DEF may now be written
paa + qbl3 + rcy = 0,
which shows that the coordinates of the line at infinity are
p = q = r, or X/a = //b = v/c.
73. To find the tangential equation of a circle.
Let
p2 = X2 + ~2 + 2 2 _ 2/Av cos A - 2Xcos B - 2X/ cos C,