some Properties of Tangents to Circles; 373 
is vertex both in BFT and ACK. And this base, being * per- 
pendicular to the right line through E, G, and V, is RL. 
Cor. 2. If ABCF be a trapezium ; one of those circles, as 
BFT, unless it coincide with ACK, includes it, and without 
contact. 
Cor. 3. If ABCF be a trapezium, either inscribed or not, 
having two points of concourse, L and R; the part NO, of the 
parallel to LR through V, intercepted between two opposite 
sides, is X bisected in V. So between AB and FC. 
Cor. 4. Fig. 13; the same as fig. 12, so far as the letters 
are common. If ABCF be a trapezium, inscribed or not, 
having two points of concourse L and R, and if FB and AC 
be the diagonals and V their mutual intersection, VD a per- 
pendicular on LR, and right lines be drawn from D to F, B, 
A, C; the angles BDF and ADC are § bisected by DV. For 
the chord bisected in V is the tangent-chord of D, in each of 
the two circles of the proposition. 
Cor. 5. Fig. 13. If FB and AC, produced, cut LR in I and K ; 
then FI is to BI as FV to VB, and AK to CK as AV to VC. || 
Cor. 6 . Fig. 13. The connecting line of a trapezium, in- 
scribed or not, is ^ divided harmonically in the four points 
L, I, R, K, in which it is cut by the produced sides and diago- 
nals. For the demonstration, of the corollary here referred 
to, may (by the present proposition and its fifth corollary) 
be applied to trapeziums not inscribed.** 
Cor. 7. Fig. 6 . Hence, if L and R be the points of concourse 
* Cor. 1 to prop. I. f Cor. 6 to prop. VI. t Cor.’ z to prop. VI. 
j Cor. 3 to prop. I. || Cor. i to prop. VI. ^ Cor. 3 to prop. X. 
** This sixth corollary ’is proved differently in Hamilton’s Stereography, B.III. 
Sect. I. Lemma 22. The remainder of that lemma is the present Cor. 5. 
JUDCCCXIV. 3 C 
