some Properties of Tangents to Circles ; &c. 
S 5 S 
INTRODUCTORY PROPOSITIONS. 
So?ne Properties (^Tangents to Circles; and (^Trapeziums, 
inscribed in Circles, and 7 ion-inscrihed, 
Definitions, for the Introductory Propositions, 
Def. 1. The tangent-chord, of a given point without a circle, 
is the chord from whose extremities tangents meet in the 
given point. 
Def. 2. A vertex to a base is any point, within a circle, 
through which are drawn chords : and the right line joining 
the points to which those are the tangent-chords, is the base 
to that vertex, 
Def. 3. The mean point is a point in the diameter perpen- 
dicular to such base : and its distance from the base is a mean 
proportional between the distances of the extremities of that 
diameter, from the base. 
Thus (Fig. 1): If the chords PQ, GH, pass through V 
within a circle, and if PD, QD, GL, HL, be tangents, and 
LD be drawn ; PQ is (def. 1 ) the tangent-chord of D, GH 
that of L ; and (def. 2) V is the vertex to the base LD. If a 
diameter F'B produced cut LD perpendicularly in D, and if, in 
DF, be taken DM equal to DP, therefore* a mean proportional 
betw'een DF and DB, M is (def. 3) mean point, f 
Prop. I. Fig. 1, 2. In a given circle, one vertex has only 
* in. Eucl. prop. 36, and VI. 14. Simson’s edition, 1762, is referred to 
throughout. 
f Other Definitions after Prop. VII. 
