(17.) The two sides of a triangle are, together, greater than the double of the straight line which joins the vertex and the bisection of the base.

(18.) If in the sides of a square, at equal distances from the four angles, four other points be taken, one in each side, the figure contained by the straight lines which join them shall also be a square.

(19.) If the sides of an equilateral and equiangular pentagon be produced to meet, the angles formed by these lines are, together, equal to two right angles.

(20.) If the sides of an equilateral and equiangular hexagon be produced to meet, the angles formed by these lines are, together, equal to four right angles.

(21.) If squares be described on the three sides of a right-angled triangle, and the extremities of the adjacent sides be joined, the triangles so formed are equal to the given triangle, and to each other.

(22.) If squares be described on the hypothenuse and sides of a rightangled triangle, and the extremities of the sides of the former, and the adjacent sides of the others, be joined, the sum of the squares of the lines joining them will be equal to five times the square of the hypothenuse.

(23.) To bisect a triangle by a line drawn parallel to one of its sides. (24.) To divide a circle into any number of concentric equal annuli.

(25.) To inscribe a square in a given semicircle.

(26.) If in a right-angled triangle a perpendicular be drawn from the right angle to the hypothenuse, and circles inscribed in the triangles on each side of it, their diameters will be to each other as the subtending sides of the rightangled triangle.

(27.) If on one side of an equilateral triangle, as a diameter, a semicircle be described, and from the opposite angle two straight lines be drawn to trisect that side, these lines produced will trisect the semi-circumference.

(28.) Draw straight lines across the angles of a given square, so as to form an equilateral and equiangular octagon.

(29.) The square of the side of an equilateral triangle, inscribed in a circle, is equal to three times the square of the radius.

(30.) To draw straight lines from the extremities of a chord to a point in the circumference of the circle, so that their sum shall be equal to a given line.

(31.) In a given triangle to inscribe a rectangle of a given magnitude. (32.) Given the perimeter of a right-angled triangle, and the perpendicular from the right angle upon the hypothenuse, to construct the triangle.

(33.) Describe a circle touching a given straight line, and also passing through two given points.

(34.) In an isosceles triangle to inscribe three circles, touching each other, and each touching two of the three sides of the triangle.

GEOMETRY OF PLANES.

DEFINITIONS.

1. A PLANE is a surface in which, if any two points be taken, the straight line which joins these points will be wholly in that surface.

2. A straight line is said to be perpendicular to a plane, when it is perpendicular to all the straight lines in the plane which pass through the point in which it meets the plane.

This point is called the foot of the perpendicular.

3. The inclination of a straight line to a plane, is the acute angle contained by the straight line, and another straight line drawn from the point in which the first meets the plane, to the point in which a perpendicular to the plane, drawn from any point in the first line, meets the plane.

4. A straight line is said to be parallel, to a plane when it cannot meet the plane, to whatever distance both be produced.

5. It will be proved in Prop. 2, that the common intersection of two planes is a straight line; this being premised,

The angle contained by two planes, which cut one another, is measured by the angle contained by two straight lines drawn from any point in the common intersection of the planes perpendicular to it, one in each of the planes. This angle may be acute, right, or obtuse.

If it be a right angle, the planes are said to be perpendicular to each other. 6. Two planes are parallel to each other, when they cannot meet, to whatever distance both be produced.

PROP. 1.

A straight line cannot be partly in a plane, and partly out of it.

For, by def. (1), when a straight line has two points common to a plane, it lies wholly in that piane.

PROP. II.

If two planes cut each other, their common intersection is a straight line.

PROP. III.

Any number of planes may be drawn through the same straight line. For let a plane, drawn through a straight line, be conceived to revolve round the straight line as an axis. Then the different positions assumed by the revolving plane will be those of different planes drawn through the straight line.

PROP. IV.

One plane, and one plane only, can be drawn,

1o. Through a straight line, and a point not situated in the given line. 2o. Through three points which are not in the same straight line.

3o. Through two straight lines which intersect each other.

4o Through two parallel straight lines.

1. For if a plane be drawn through the given line, and be conceived to revolve round it as an axis, it must in the course of a complete revolution pass through the given point, and so assume the position enounced in 1o.

Also, one plane only can answer these conditions, for if we suppose a second plane passing through the same straight line and point, it must have at least two intersections with the first, which is impossible.

2. Join two of the points, this case is then reduced to the last.

3. Take a point in each of the lines which is not the point of intersection, join these two points; the case is now the same as the two former.

4. Parallel straight lines are, by their definition, in the same plane, and, by the first case, one plane only can be drawn through either of them, and a point assumed in the other.

Cor. Hence, the position of a plane is determined by,

J. A straight line, and a point not in the given straight line.
2. A triangle, or three points not in the same straight line.
3. Two straight lines which intersect each other.

4. Two parallel straight lines.

PROP. V.

If a straight line be perpendicular to two other straight lines which intersect at its foot in a plane, it will be perpendicular to every other straight line drawn through its foot in the same plane, and will therefore be perpendicular to the plane.

Let XZ be a plane, and let the straight line PQ be perpendicular to the two straight lines AB, CD which intersect in Q in the plane XZ. Draw any straight line EF through Q;

Then PQ will be perpendicular to EF. Draw through any point K in QF a straight line

GH, such, that GK = KH.

Join P, G; P, K; P, H;

Then, since GH, the base of the A GQH, is bisected in K;

X

P

E K

Q

SA

...(1)

Similarly, since GH, the base of A GPH, is bisected in K;

... GP + HP = 2GK2PK'.

But the angles PQG, PQH, are right angles, .. the above beconies,

In like manner, it may be proved that PQ is at right angles to every other straight line passing through Q in the plane XZ.

PROP. VI.

A perpendicular is the shortest line which can be drawn from any point to

a plane.

Let PQ be perpendicular to the plane XZ;

From P draw any other straight line PK to the plane XZ;

In the plane XZ draw the straight line QK, joining the points Q, K.

Then, since the line PQ is perpendicular to the plane XZ, the angle PQK is a right angle; and .. PQ is less than any other line PK. (Geom. Theor. xxi.)

Cor. 1. Hence, oblique lines equally distant from the perpendicular are equal, and, if two oblique lines be unequally distant from the perpendicular, the more distant is the larger.

That is, if QG, QH, QK,

are all equal,

then PG, PH, PK,
are all equal; and
if QH be greater than QG, then PH is greater than
PG.

Cor. 2. A perpendicular measures the distance of any point from a plane. The distance of one point from another is measured by the straight line joining them, because this is the shortest line which can be drawn from one point to another. So also, the distance from a point to a line, is measured by a perpendicular, because this line is the shortest that can be drawn from the point to the line. In like manner, the distance from a point to a plane, must be measured by a perpendicular drawn from that point to the plane, because this is the shortest line that can be drawn from the point to the plane.

PROP. VII.

Let PQ be a perpendicular on the plane XZ, and GH a straight line in that plane; if from Q, the foot of the perpendicular, QK be drawn perpendicular to GH, and P, K, be joined; then PK will be perpendicular to GH.

Take KG = KH, join P, G; P, H; Q, G ; Q, H; ·.· KG = KH, and KQ common to the triangles GQK, HQK, and angle GKQ = angle HKQ, each being a right angle.

QG = QH

Hence, the two triangles GKP, HKP, have the two sides GK, KP, equal to the two sides HK, KP, and the remaining side GP, equal to the remaining

side HP.

P

X

.. Angle GKP = angle HKP, and .. each of them is a right angle. Cor. GH is perpendicular to the plane PQK, for GH is perpendicular to each of the two straight lines KP, KQ.

REMARK. The two straight lines PQ, GH, present an example of two straight lines which do not meet, because they are not situated in the same plane.

The shortest distance between these two lines is the straight line QK, which is perpendicular to each of them.

For, join any two other points, as P, G;

The two lines PQ, GH, although not situated in the same plane, are considered to form a right angle with each other. For PQ, and a straight line drawn through any point in PQ parallel to GH, would form a right angle.

In like manner, PG, and QK, which represent any two straight lines not situated in the same plane, are considered to form with each other the same angle which PG would make with any parallel to QK, drawn through a point in PG,

PROP. VIII.

If two straight lines be perpendicular to the same plane, they will be parallel

to each other.

Let each of the straight lines PQ, GH, be perpendicular to the plane XZ.

Then, PQ will be parallel to GH. In the plane XZ draw the straight line QH, joining the points Q,H.

Then, since PQ, GH, are perpendicular to the plane XZ; they are perpendicular to the straight line QH in that plane; and, since PQ, GH, are both perpendicular to the same line QH, they are parallel to each other. (Geom. theor. 13, cor.)