PROPERTIES of the CIRCLE. 23 



In the fame way, it is demonftrated, that if from any two 

 points p, q, 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 n. 



But if from any two points V, W, in PQ^ produced, equally 

 diflant from the centre O, lines drawn perpendicular to any dia- 

 meter B r, pafhng through any point of contacl B, fall beyond 

 its extremities B, r, 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 paiTing through the points of contacl 



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 p and q, to 

 lines touching the circle in the points A, B, C, &c. are toge- 

 ther equal to a multiple of the radius by 2 n. 



Cor. 2. The fum of perpendiculars drawn from P, Q^ or p, q, 

 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 circumfcribing the 

 circle from any point within the fame regular figure. 



r AP +BF-J-CP+, &c. r c , „. , 



<~<or. 3. . — fum or the perpendiculars 



drawn from P to right lines touching the circle in the points A, 



B, C, &c. d denoting the diameter. 



Or a third 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 



