some Properties of Tangents to Circles ; &c. 367 
is the base to which V, the intersection of AC with BF, is 
vertex in the circle ABCF. 
Dem. Let ABC be the near ang^le, AFC the remote, E the 
centre, D the dividing point, and A in LF. Draw ED, DA, DC, 
DB, DF. Draw or conceive the circles* ABDL &c. Then, by 
ABDL, the triangle LDA is-f* similar to LFR; and, by CBDR, 
RDC to RFL : whence the angles LDA, RDC, are equal, 
and ADC is bisected by DE. Further. If BF be a diameter, 
it coincides with ED ; which J bisects the tangent-chord of D. 
Let it be not a diameter. By the circle ABDL, the angle 
LDB has the supplement LAB, and is equal to BAF ; and, by 
FADR, RDF is equal to RAF or BAF: whence LDB, RDF, 
are equal ; also LDF, RDB ; and BDF is bisected by DE. 
Therefore the tangent-chord of D is § bisected by AC and 
BF, in V. But ED bisects it : whence V is its intersection with 
ED, therefore II the vertex to LR. Q. E.D, 
Cor. 1 . Fig. 5. If two sides, as GS and HT, be parallel ; a line 
KC parallel to them is the base. For, let GH and ST meet in 
K,GT and SH in I. Then one diameter LR bisects GS and HT 
perpendicularly, and, produced, bisects the angle GKT or 
HKS; whence GT and SH bisect^ the tangent-chord of K in 
I ; which is perpendicular to LK or LR, and so is parallel to 
GS and HT ; whence KC is ** base to the vertex I. Also GH 
and ST are equal. 
Cor. 2. Fig. 9. DAEC, DBEF, are circles; except BF be 
a diameter of the inscribing circle. For the rectangle DVE 
* Cor. 3 to prop. IX. 
J Dem. of prop. I. 
fj Cor. I to prop. I. 
** Cor. 1 to prop. I. 
3B z 
f Dem. of prop. IX. 
^ Cor. 3 to prop. I. 
^ Cor. 3 to prop. I. 
