the triangle DCQ; also because BD is equal to DC, therefore the triangle BDA is equal to the triangle CDA; therefore, taking the former equal triangles from the latter, there remains the triangle DPA equal to the triangle DQA. But since PG is greater than GQ, the triangle APG is greater than AGQ, and the triangle GPD is greater than GQP; therefore, the whole triangle DPA is greater than the whole triangle DQA; but it has been shown that the triangle DPA is equal to the triangle DQA; and it is also greater, which is absurd: hence neither of the two lines PG, GQ, can be greater than the other; therefore PG must be equal to GQ.

2. The two sides of a triangle are together greater than the double of the line joining the vertex and the middle of the base.

Let ABC be a triangle, and AD the straight line joining the vertex A, and the middle D, of the base BC; then AB and AC are together greater than twice AD. Produce AD to Q making DQ equal to AD; and join BQ.

B

Q

D

Since (hyp.)BD is equal to DC, and (constr.) DQ is equal to AD, and (E. 1. 15.) the angle BDQ is equal to ADC; therefore (E. 1. 4.) BQ is equal to AC. But (E. I. 20.) A B, BQ are together greater than AQ; but AC has been proved to be equal to BQ, and AQ (by const.) is the double of AD; therefore AB, AC are together greater than the double of AD.

3. If either diagonal of a parallelogram be equal to one of the sides, the other diagonal will be greater than any side of the parallelogram.

4. A trapezium, having two of its sides parallel, is equal to half of a rectangle between the same

parallels, and having its base equal to the sum of the two parallel sides of the trapezium.

5. If the side of a square be equal to the diagonal of another square, the former square is double of the latter.

6. The two sides of a triangle are together greater than the double of the line drawn from the vertex to the base, bisecting the vertical angle.

7. 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.

8. If four straight lines cut each other, without including a space, but so as to make three internal angles, towards the same parts, together less than four right angles, the two lines which are not joined shall meet, if produced.

9. If two opposite sides of a parallelogram be bisected, and lines be drawn from these two points of bisection to the opposite angles, the lines will be parallel, and will trisect the diagonal.

10. In Prob. 4. No. 1. the sum of the lines CE and ED is less than the sum of any other two lines which can be drawn from the given points c and D to meet the line AH.

11. The perimeter of an isosceles triangle is less than that of any other equal triangle on the same base.

12. If from the angular points of the squares described upon the sides of a right-angled triangle perpendiculars be let fall upon the hypotenuse produced, they will cut off equal segments, and the perpendiculars will together be equal to the hypotenuse.

13. If a line PQ be drawn parallel to the base BC of a triangle ABC (see fig. Theo. 1. No. 2.) through

the point & where the lines CG and BG bisecting the angles of the base intersect, the line PQ will be equal to the sum of the lines BP and CQ.

14. The difference of the angles at the base of a triangle is double the angle contained by a line drawn from the vertex perpendicular to the base, and another line bisecting the angle at the vertex.

15. If in two opposite sides of a parallelogram two points be taken, one in each of those sides equidistant from two opposite angles of the figure, and if two other points be likewise taken, in the two other opposite sides equidistant from the same two angles, the figure contained by the straight lines joining the four points so taken shall be a parallelogram.

16. The area of a trapezium is half that of a rectangle, whose base is one of the diagonals of the trapezium, and altitude the sum of the perpendiculars let fall upon the diagonal.

17. The lines which bisect the angles of a parallelogram, form a rectangle.

18. Two sides of a triangle being given, its area will be greatest when the given sides contain a right angle.

19. If the two exterior angles at the base of a triangle be bisected, and the bisecting lines produced until they intersect, the line drawn from this point to the vertical angle will bisect it.

I.

20. In the figure E. 1. 47. prove that, if BG and CH be joined, these lines will be parallel.

EXERCISES ON BOOK II.

1. THEOREM. In any triangle, if a line be drawn from the vertex bisecting the base, the sum of the

squares of the two sides of the triangle is double the sum of the squares of the bisecting line and of half the base.

C

Let ABC be a triangle; and from the vertex c let CD be drawn to the middle, D, of the base; then the squares of AC, BC are together double the squares of DC, AD.

From c draw CE dicular to AB.

Then (E. n. 12.) the square of AC is greater than the squares of DC, AD, by twice the rectangle AD, DE; and (E. II. 13.) the square of BC is less than the squares of DC, BD by twice the rectangle BD, DE; therefore the squares of AC, BC are equal to twice the square of DC, and twice the square of AD, since AD is equal to BD.

(E. II. 12.)

Or thus,

AC2 = DC2 + AD2 + 2 AD. DE, and (E. II. 13.) BC2=DC2+ BD2 — 2BD. DE =DC2 + AD2 — 2 AD. DE,

.. AC2 + BC2 = 2DC2+2 AD2.

2. The square upon the whole line is equal to four times the square upon half the line.

3. The rectangle under any two lines is equal to three times the rectangle under either of them and a third of the other.

4. If two lines be divided, each into any number of parts, the rectangle contained by the two lines, is equal to the sum of the rectangles contained by the several parts of the one and the several parts of the other.

5. In any triangle if a line be drawn from the vertex at right angles to the base, the difference of the squares of the sides is equal to the difference of the squares of the segments of the base.

6. If from one of the acute angles of a rightangled triangle, a line be drawn to the opposite side, the squares of that side and the line so drawn are together equal to the squares of the segment adjacent to the right angle and of the hypotenuse.

7. In any isosceles triangle, if a line be drawn from the vertex to any point in the base, the square upon this line, together with the rectangle contained by the segments of the base, is equal to the square upon either of the equal sides.

8. The rectangle contained by the sum and the difference of two lines is equal to the difference of their squares.

9. If a line be divided into five equal parts, the square of the whole line is equal to the square of the line which is made up of four of those parts, together with the square of the line which is made up of three of those parts.

10. The squares of the sides of a parallelogram are together equal to the squares of its diameters taken together.

11. If from the three angles of a triangle lines be drawn to the points of bisection of the opposite sides, four times the sum of the squares of these lines is equal to three times the sum of the squares of the sides of the triangle.

12. If from any point within a rectangle lines be drawn to, the angular points, the sums of the squares of those which are drawn to the opposite angles are equal.

13. The square of the base of an isosceles triangle is double the rectangle contained by either side and by the line intercepted between the perpendicular let fall upon it from the opposite angle, and the extremity of the base.

14. If two sides of a trapezium be parallel to