some Properties of Tangents to Circles ; ^c, 37 1 
Cor. 1. If the two circles coincide, ABCF becomes an inscribed 
trapezium : and thus is obtained a separate proof of the tenth 
proposition. * * * § 
Cor. 2. If, in any number -f* of circles having a common 
vertex and base, chords through the vertex be made diagonals 
of any trapeziums, inscribed or not; such base is the locus of 
all the points of concourse. 
Prop. Xil. Fig. 12. If a non-inscribed J trapezium have two 
points of concourse ; the connecting line is the base to which 
the mutual intersection of the diagonals is vertex in two circles, 
in each of which one diagonal is a chord. 
Thus : if ABCF have L and R its points of concourse ; LR 
is the base to which V, the intersection of AC with BF, is vertex 
in the circles ACK and BFT. 
Dem. Let VD be the perpendicular on LR. Let ABC be 
the near angle, AFC the remote, and A between L and F. 
Through V draw NO parallel to LR, and indefinite. And let 
AC be first supposed to cut NO, and C to be nearer to LR" 
than A is. Let lines bisecting BF and AC perpendicularly 
cut DV in E and G. With the centres E and G describe the 
circles BFT and ACK. In both, V is vertex to the base LR. 
For two circles, through F and A respectively, can § he de- 
scribed, each having V its vertex to LR. If these do not pass 
through B and C respectively, let them cut FB and AC in 
other points, X and Y ; and complete the trapezium AXYL’. 
* The former proof is retained, as independent of harmonical proportion (involved 
in this proposition), and of the deduction by the coincidence. 
I See Cor. 4 to prop. I. ; or Cor. 5 to prop. VI. 
j That is, one which cannot stand inscribed in a circle. 
§ Cor. 5 to prop. VI. 
