137. Given two circles. Find the locus of a point which moves so that the difference of the squares on the tangents from it to the two circles shall be constant.

NOTE. When the difference is 0, the locus is called the radical axis.

138. Find the locus of a point from any position of which two given circles subtend the same

larger circle, is internally tangent to the larger one and rolls on its circumference.

Find the locus of any given point P on the circumference of the smaller circle.

Query.If segments of the rolling circumference be of different colors, what will be the effect of a very rapid rolling of the smaller circle on the larger one ?

NOTE. This is a special case of a general problem; viz. Find the locus of any point in the plane of a circle, as the latter rolls on the circumference of another circle.

FIG. 236.

P

The general case is better treated by the methods of Algebraic Geometry.

141. Find the locus of a point, the distances of which from two given straight lines have a fixed ratio.

142. Find the locus of a point which moves so that the tangents from it to two given circles are equal.

143. Find the locus of a point which moves so that the sum of its distances from two vertices of an equilateral triangle shall equal its distance from the third.

144. Find the locus of a point which moves so that the sum of the squares of the distances from two given points is fixed. The same for three given points.

145. Find the locus of a point which moves so that the difference of the squares of its distances from two fixed points is constant.

146. Within a circle two chords at right angles to each other intersect in a fixed point. Find the locus of the middle points of the sides of the quadrangle of which the perpendicular chords are the diagonals.

SOLID AND SPHERICAL GEOMETRY.

CHAPTER IX.

The first eight chapters of this work have dealt chiefly with the relations of figures in a single plane. The remainder, making use of what has been developed in the Plane Geometry, will not be confined to a plane.

105. THEOREM. The intersection of two planes is a straight line.

A plane has been defined as a surface such that if any two points in it be joined by a straight line, the line will be wholly in the surface.

If two planes intersect, they will have more than one point in common. Any two of these points determine a straight line, which straight line will lie in both planes, hence must be common to both and so be their intersection. Q. E. D. Planes are usually represented by a quadrangle which is a part of the surface, and the quadrangle is usually

taken in the form of a parallelogram, and is designated

as the plane CE, for instance.

Exercises.

mine a plane.

1. Show that a straight line and a point deter

2. Show that three points determine a plane.

3. Show that a pair of intersecting straight lines determine a plane.

106. When two planes intersect, they form four diedral angles.

A diedral angle between two planes is the amount of rotation that one plane would have to undergo about the line of intersection as an axis, to coincide with the other. The measure of a diedral angle is the plane angle formed by two lines, one in each plane, perpendicular to the line of intersection at the same point.

Let MN and QS represent two planes, IJ the line of intersection, B any point in that line, BA a line in the plane MN 1 to IJ, and BC a line in the plane QS to IJ.

The plane ABC is

the measure of the diedral

M-IJ-S.

M

J

-S

H

B

N

FIG. 238.

The vertical plane HBK is the measure of the diedral, vertical to the one measured by ABC.

The plane angle ABH, the supplement of ABC, is the measure of the diedral M-IJ-Q.

CBK, the vertical of ABH, is the measure of S-IJ-N, the vertical of M-IJ-Q. Hence the

THEOREM.

Vertical diedrals are equal.

107. Diedrals are acute, right, or obtuse according as the measuring plane angle is acute, right, or obtuse.

B

FIG. 239.

If a diedral be 90°, the line BA will be perpendicular to both BI and BC, which are lines on the plane QS.

THEOREM. If a line be perpendicular to two lines of a plane at their intersection, it will be perpendicular to any line of the plane passing through its foot.

Let BA be perpendicular to BC and BH. Let BP be any line of the plane QS passing through B.

Analysis. If BP is perpendicular to AK and only under that condition, we know that oblique lines drawn from any point of BP to points equally distant from the foot of the perpendicular will be equal.

H

FIG. 240.

Demonstration.- Draw HC, a straight line in the plane QS intersecting the three lines of the plane at H, P, and C. Lay off BK = BA and join the points II, P, and C with A and K.

N