7. Any point is taken on an arc of a circle. Prove that the distance of this point from the chord of the arc is a mean proportional between the distances of the same point from the tangents at the extremities of the arc.

8. If a tangent be drawn to any circle at any point, prove that the radius of the circle is a mean proportional between the lengths of this tangent intercepted between the point of contact and the tangents at the extremities of any diameter.

9. ABC is a triangle, and M is the middle point of the side BC. I is the point in which the inscribed circle touches BC, and H and K are the points in which the perpendicular from A upon BC, and the line bisecting the angle at A meet BC. Prove that

10. Given an angle of a triangle, the perpendicular from this angle upon the opposite side, and the sum or difference of the remaining sides. Construct the triangle.

II. Through two given points draw two straight lines intersecting on a given circle, and such that the chord joining the other two points of intersection with the circle may be parallel to the straight line joining the two given points.

12. Determine a point such that its distances from the three sides of a given triangle shall be proportional to three given straight lines.

13. Divide a given straight line into three parts, such that the first and second may be in a certain given ratio, and the second and third in another given ratio.

14. If from the angles of an equilateral triangle perpendiculars be drawn upon any diameter of the circumscribing circle, prove that the perpendicular which falls upon one side of that diameter will be equal to the sum of the two perpendiculars which fall upon the other side.

15. In a given square inscribe four equal circles touching one another.

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BOOK VII.

ON PLANES, AND LINES IN SPACE.

SECTION I.-MISCELLANEOUS PROPOSITIONS.

PROPOSITION 1.

It is always possible to find one plane, and only one plane, containing a given straight line and a given point not in that straight line.

A

Let BC be the given straight line and A the given point

B

Fig. 1.

G

F

E

D

C

not situated in the straight line BC, then it shall be always possible to find one plane, and only one plane, containing the straight line BC and the point A.

Suppose a plane of indefinite extent to revolve round the straight line BC, then since

in the course of its revolution such a plane must pass through every point in space, therefore there must be some one position of this plane in which it passes through the given point A.

That is, one plane may always be found containing the given straight line BC and the given point A.

Also, no more than one such plane can be found.

If possible, let ABCD and ABCE be two non-coincident planes, each of which contains the straight line BC and the point A.

Take any point F in BC.

Because each of the points A and F is situated in the

plane ABCD, therefore a straight line AGF may be drawn from A to F, and situated in the plane ABCD (Def. 9).

Similarly, a straight line AHF may be drawn from A to F, and situated in the plane ABCE.

Therefore the two non-coincident straight lines AGF and AHF intersect in two points A and F,

which is impossible (Ax. 1);

therefore the planes ABCD and ABCE must coincide in every point.

Corollary 1.-One plane, and only one plane, may always be found containing three given points not situated in the same straight line.

Corollary 2.-One plane, and only one plane, may always be found containing two given intersecting straight lines.

PROPOSITION 2.

It is always possible to find one plane, and only one plane, containing two given parallel straight lines.

Let AB and CD be the two given parallel straight lines, then it shall be always possible to find one plane, and only one plane, containing both of the straight lines AB and CD.

Because AB and CD are parallel straight lines, therefore they are situated in one plane (Def. 27);

Fig. 2.

A

C

E

that is, one plane may always be found containing both

AB and CD.

Also no more than one such plane can be found.

For let E be any point in CD.

D

Because E is a point not situated in the straight line AB, therefore only one plane can be found containing the straight line AB and the point E (Bk. VII. Prop. 1);

R 2

therefore only one plane can be found containing the two given parallel straight lines AB and CD.

Note. It follows from the two preceding propositions that a plane is completely determined in each of the three following cases:

1. When it contains a given straight line and a given point not in that line.

2. When it contains three given points not situated in the same straight line.

3. When it contains each of two given intersecting or parallel straight lines.

PROPOSITION 3.

Through any given point it is always possible to draw one straight line, and one straight line only, parallel to any given straight line not containing the given point.

Let A be the given point, and BC the given straight line not containing the point A, then it shall be always possible to draw one straight line, and one straight line only, through the point A parallel to the straight line BC.

Fig. 3.

B

A

C

D

Because the point A is not situated upon the straight line

BC, therefore one plane, and one plane only, may always be found containing the straight line BC and the point A.

Because (Ax. 4 and Bk. III. Prob. 6) it is always possible to draw one straight line, and one straight line only, in the plane containing the straight line BC and the point A, and parallel to BC, let this straight line be AD; therefore AD will be drawn through the given point A, parallel to the given straight line BC, and will be the only straight line that can be so drawn.

PROPOSITION 4.

If two planes cut one another their line of intersection shall be a straight line.

Fig. 4.

Let ABC and ABD be two planes which cut one another, and let A and B be two points in their line of intersection, then if AB be joined, every point common to both of the planes shall be situated in the straight line AB or in AB produced.

If possible let P be a

P

A

B

D

point in the line of intersection of these planes and not situated in the straight line AB or that line produced.

Because A, P, and B are three points not situated in the same straight line, therefore no more than one plane can be found which contains the points A, P, and B.

Because A, P, and B are three points situated in the line of intersection of the two planes ABC and ABD, therefore there are two planes ABC and ABD, each of which contains the three points A, P, and B.

But we have before proved that only one plane can be found containing these three points,

which is impossible;

therefore every point in the line of intersection of the planes ABC and ABD must be in the straight line AB; that is, the line of intersection of these planes is a straight line.

DEFINITIONS.

79-A straight line and a plane are said to be parallel to one another when they cannot meet, however far they may be produced in any direction whatever.