*(33) From Euc. II. 7, prove that the sum of the squares on two straight lines is never less than twice their rectangle.

*(34) On the base of a given triangle construct an oblong equal to one fourth of the given triangle.

(35) Find the number of sides in an equiangular polygon which has four angles together equal to seven right angles.

*

(36) If from the extremities of any chord in a circle, lines be drawn to any point in the diameter to which it is parallel, the sum of their squares is equal to the sum of the squares on the segments of the diameter.

*(37) The centre of a circle being given, find two opposite points in the given circumference by means of a pair of compasses only.

* (38) If the sides of a triangle be as 2, 4, 5, show whether it will be acute or obtuse angled.

* (39) In any triangle the squares on the two sides are together double of the squares on half the base and on the line joining its middle point with the opposite angle.

Let A B C be a triangle, having the base B C. Bisect B C in D; from A draw A E perpendicular to B C, and join A D.

Then the square on A B is less than the squares on B D and A D by twice the rectangle B D, D E (II. 13).

And the square on AC is greater than the squares on CD and A D by twice the rectangle C D, D E (II. 12); that is, than the squares on B D and A D by twice the rectangle B D, D E. Therefore the squares on A B and A C are together double of the squares on B D and A D.

* (40) ABC is a triangle right-angled at A, and BE, CF bisect the opposite sides respectively. Show that four times

the sum of the squares on BE and CF is equal to five times the square on B C.

* (41) If two adjacent sides of a parallelogram be parallel to two adjacent sides of another parallelogram, the other sides will also be parallel.

(42) Describe an obtuse-angled triangle, such that the square on the largest side may be equal to three times the square on either of the equal sides.

(43) The opposite sides of any equiangular rectilineal figure must be parallel, if the number of sides be even.

(44) Taking the figure of Euclid I. 47, join G H, K E, and DF; then prove that the three triangles GA H, KC E, and DBF are equal, each of them being equal to the triangle A B C.

(45) Trisect a given straight line.

(We give the construction of this problem, leaving the demonstration to the pupil.)

Let A B be the given straight line; on A B describe the equilateral triangle A B C. Bisect the angles at A and B by the straight lines A D, B D, meeting within the triangle at D. From D draw D E parallel to CA and D F parallel to C B, meeting A B in E and F. The given straight line A B is trisected in the points E and F.

* (46) Given a line divided into two parts, as in Proposition II of Book II.; show that the squares on the whole line and one part are equal to three times the square on the other part. (See figure to Euclid II., 11.)

The squares on AB, BH are together equal to twice the rectangle A B, B H, together with the square on A H (II. 7). But the rectangle A B, B H is equal to the square on A H (II. 11).

Therefore twice the rectangle A B, B H is equal to twice the square on A H.

Therefore the squares on AB, BH are together equal to three times the square on A H.

* (47) If D, E, F are points taken in the sides B C, C a, a b

of an equilateral triangle A B C, so that B D=CE=AF, show that the triangle D E F is equilateral.

*(48) If A B C D be a parallelogram, and P any point in the diagonal A C, show that the triangles P B C and P D C are equal.

*(49) From any point P lines are drawn to the angles of a rectangle A B C D; show that the squares on PA, PC are together equal to the squares on P B, P D.

*(50) The base angle of an isosceles triangle is one fourth of the vertical angle, and from it a line is drawn perpendicular to the base to meet the opposite side produced; show that the part produced, the perpendicular, and the remaining side will form an equilateral triangle.

B

draw B D at right Then D B A shall

Let A B C be the isosceles triangle. Let the angle ABC be one fourth of the angle B AC. From B angles to B C, meeting C A produced in D. be an equilateral triangle.

The angle B A C may be proved equal to two-thirds of two right angles. Also each of the angles of the triangle D BA may be proved equal to one-third of two right angles. Hence the triangle is equiangular, and therefore equilateral. (We leave the full demonstration to be worked out by the student.)

*(51) AOB is a quadrant of a circle, whose centre is O, and from any point P in its arc, P M perpendicular to O A or O B meets the radius which bisects the angle A O B in C; show that the squares on PM, CM. are together equal to the square on the radius.

Let O F be the radius which bisects the angle AO B. Take

P between F and A.

Then P M must be drawn perpendicular to O B. (If P be between F and B, then P M must be drawn perpendicular to O A.)

Join O P. Now prove that M O is equal to M C.

Then from the right-angled triangle P M O prove that the squares on P M and C M are equal to the square on P O.

*(52) If straight lines be drawn from each angle of a triangle to bisect the opposite side, four times the sum of the squares on these lines is equal to three times the sum of the squares on the sides.

(Note.-Apply Euc. II. 12, 13.)

*(53) The two triangles, formed by drawing lines from any point within a parallelogram to the extremities of two opposite sides, are together half the parallelogram.

*(54) The sum of the squares on the sides of any quadrilateral figure is equal to the sum of the squares on its diagonals, together with four times the square on the line joining the middle points of the diagonals.

In solving this problem it may be assumed that in every triangle the sum of the squares on two of the sides is equal to twice the square on half the base, together with twice the square on the line drawn from the vertex to the middle point of the base. (See No. 39.)

MENSURATION.

Mensuration (Lat. mensura, a measure) is the art of measuring the dimensions of bodies and figures.

The following tables are used in the measurement of surfaces.

A mile contains 1,760 yards, or 5,280 feet.

The following table of linear measure is used in the measure

I square mile.

An acre contains 10 square chains, or 100,000 square links.

I. To find the Area of a Square, Rectangle, or other Parallelogram.

Rule.-Multiply the base by the perpendicular height.

Proof. CASE 1.-Let A B C D be a rectangle, of which the