Must these loci intersect? If the common tangent and m n do not intersect, where must point P lie? What is the locus of the center of a circle tangent to two parallel lines?

The above discussion is called the analysis of the probHaving discovered the solution, we now give the direct construction as follows:

lem.

Given: (1) Line m n.

(2) Circle A.

(3) Point P in circumference of A.

Required: To draw a tangent to OA at P and to the line m n.

Construction: (1) Draw A P and produce it.

(2) Draw tangent to A at P and produce it until mn is intersected (if possible).

(3) Letter point of intersection of tangent and m n, x, and then bisect / m x P and produce the bisector until A P produced is intersected. Letter point of intersection B. Then B is the center of the required O.

Proof: (1) A P produced is the locus of the center of a tangent to OA at P.

(2) Px is 1 to A B.

(3) The bisector of

[Ex. 96.]

[§ 148.]

m x P is the locus of the center

of a tangent to m n and x P. [Ex. 84 and Ex. 98.]

OB is the required O.

[Pupil will give solution when common tangent and m n are ||; also when A P and m n are ||; also when A P produced (beyond P) will not cut m n.]

107. Construct a

tangent to a given line at a given

point and also tangent to a given

What have we given? (1) AO to which the required

is somewhere to be tangent. (2) A line and the point of tangency to that line.

to that line.

Again suppose the problem solved-for what purpose? Given: (1) m n the given line and A the given O. (2) Pthe point of tangency of the required (3) And B the required Again try to find two loci which intersect and thus fix B. [See Ex. 106.]

(supposed to be).

Can you get the locus of the center of a tangent to another

?

Can you get the locus of the center of a line at a given point?

tangent to a

Can you get another locus to intersect the locus found? Review the locii you have learned. See Ex. 62, etc.

Can you apply Ex. 62? (1) Is B equidis ant from the two fixed points which were given? If not, is it possible, with data given, to fix a point which you can use with P or with A? What straight lines can

[What straight line is given?

you draw? Try to fix a point by measuring from fixed points on given lines or lines which you can draw.]

Ex. 62 can be used here. But if it could not, you should try another locus, etc., etc., till you get a solution.

108. Construct a rectangle when the perimeter and the diagonal are given. This is a difficult problem and it is of much importance that the pupil fully weigh eacl question in the order given, and frequently review the ground gone over to hold in mind just what has been discovered, that it may be readily used in further investigation.

Note carefully the data given (what we know). Draw any rectangle-suppose it to be the required rectangle and study it carefully. What data are necessary to draw it? Have we sufficient data given? If not, how can we by using data given get the necessary lines and points to draw it?

How many corners of the required rectangle do data enable us to fix?

If we assume A as fixed, can we get B, or the opposite corner, C? Can we get the locus of either of these corners?

What is the locus of C? Is it a fixed distance from A? Do you know that distance?

Locus of D.

Let us try to discover another locus which will enable us to fix C.

[But why do we want to fix C? How could you draw the rectangle if it were fixed?]

Let us now construct the rectangle so far as possible.

But we must have another locus of C, or of D.

We have a and also b; ... we can get a part of them should we so desire. What use could we make of 1⁄2 the diagonal, or of 1⁄2 the perimeter? Study the supposed rectangle. You can see there another locus of C. What is it? [Circumference of a having B the center and B C the radius.] You no doubt think that we have neither point B, nor radius, B C, but there is a point in that circumference which we can fix. Produce A B till it is the length of the known perimeter. At what point does the above circumference cut A B produced? Can you not fix that point? Letter it x. Note the / CB x. What kind of ▲ would it form the sides of? How could you draw at x the third side of the A, or a line that must contain C, giving a second locus of C?

109.

Construct a with a given radius:

(1) Tangent to two given lines.

(2) Tangent to a given line and to a given O.

[Find intersection of loci.]

(3) Tangent to two given Os.

Cor. I. Pass the circumference of a through three given points. When is this impossible?

Cor. II. Find the center when an arc is given.

Problem.

169.

PROPOSITION XXIV.

To inscribe a in a given A.

170.

Cor. I. To draw a tangent to three given lines. When is this impossible?

171.

PROPOSITION XXV.

Problem. To construct on a given chord a segment which will contain a given inscribed L.

[Use any convenient point on either side of ≤ b as a cen

ter and with a for radius cut the other side of b.

Pass a cir