A sequel to the first six books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples. By John Casey.

152

A SEQUEL TO EUCLID

the Os described about these As intersect in five points A", B",
C", D", E", which are concyclic.

Dem.-Join E"B', E"D", D"C", C"B", C"C; join also D"D
and E'B", and let them produced meet in G. Now, consider the A
AB'E', it is evident the 0 described about it (Cor. 3, Prop. 12,
Book III.) will pass through the points E", B"; hence the four
points E", B', E', B" are concyclic;.'. the L GB"E"= E'B'E";
but E'B'E"= GD"E";.. L GB"E" = GD"E". Hence the
0 through the points B", D", E" passes through G.
Again, since the figure CDD"C" is a quadrilateral in a 0, the
L GDE' = D"C"C, and the L GE'D = B"C"C (III. 21);.-. L B"C"D" = GDE' + GE'D. To each add L E'GD, and
we see that the figure GD"C"B" is a quadrilateral in a 0D;
hence the G through the points B", D", E" passes through C".
In like manner it passes through A". Hence the five points
A", B", C", D", E" are concyclic.
43. If the product of the tangents, from a variable point P to
two given Os, has a given ratio to the square of the tangent from
P to a third given G coaxal with the former, the locus of P is ai
cicle of the same system.