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.

122

A SEQUTEL TO EUCLID.

radical axis of the Os ABC, A'B'C'; but since PA = PA',
being tangents to X, the point P is on this radical axis.
Hence P is the point of intersection of two given lines,
namely, the axis of similitude of X, Y, Z, and the
chord common to X and the orthogonal ( DEF;.'. P
is a given point; hence A, A', the points of contact of the
tangents from P to X, are given. Similarly, the points
B
B, B'; C, C' are given points. And we have the following construction, viz.: Describe the orthogonal circle of
X, Y, Z, and draw the three chords of intersection of this
circle with X, Y, Z respectively; and from the points
where these chords meet the axis of similitude of X, Y, Z
draw pairs of tangents to X, Y, Z; then the two circles
described through these six points of contact will be tan.
gential to X, Y, Z.
Cor. 1.-Since there are four axes of similitude of
X,Y, Z, we shall have eight circles tangential to X,Y, Z.
Cor. 2.-If we suppose one of the circles to reduce to
a point, we have the problem: '" To describe a circle
touching two given circles, and passing through a given
point." And if two of the circles reduce to points, we
have the problem: " To describe a circle touching a given
circle, and passing through two given points."