PROPERTIES of the CIRCLE. 23 
In the fame way, it is demonftrated, that if from any two 
points Pp, 7, in the diameter PQ, equally diftant from the centre 
O, perpendiculars be drawn to the lines touching the circle in 
the points A, B, C, &c. their fum is equal to a multiple of the 
diameter by 7. 
Bur if from any two points V, W, in PQ_ produced, equally 
diftant from the centre O, lines drawn perpendicular to any dia- 
meter Br, pafling through any point of contact B, fall beyond 
its extremities B, 7, the difference of the perpendiculars drawn 
from W, V, to the line touching the circle in B, is equal to the 
diameter, and fo on. ; 
So alfo, when perpendiculars from the points V, W in PQ, 
produced to the diameters pafling through the points of contact 
A, B, C, &c. do not fall beyond the extremities of any of thefe 
diameters, perpendiculars from V and W to right lines touching 
the circle in the points A, B, C, &c. are taken together equal to 
a multiple of the diameter by the number of the faid points. 
Cor. 1. Perpendiculars drawn from P and Q, or f and g, to 
lines touching the circle in the points A, B, C, &c. are toge- 
ther equal to a multiple of the radius by 22. 
Cor. 2. The fum of perpendiculars drawn from P, Q, or Ps 9 
to the fides of any regular figure circumfcribed about the circle, 
is equal to twice the fum of perpendiculars drawn to the fides of 
a regular figure of the fame number of fides circum{cribing the 
circle from any point within the fame regular figure. 
Cor. 3. (2 = fum of the perpendiculars 
drawn from P to right lines touching the circle in the points A, 
B, C, &c. d denoting the diameter. 
Or athird proportional to the diameter and the chord AP, to- 
gether with a third proportional to the diameter and the chord 
BP, together with a third proportional to the diameter and the 
chord 
