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PROPOSITION XIII. THEOREM. In every triangle, the square on the side subtending any of the acute angles is less than the squares on the sides containing that angle, by twice the rectangle contained by either of these sides and the straight line intercepted between the perpendicular, let fall upon it from the opposite angle, and the acute angle.

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B

sq. on CD.

с Let ABC be any A, having the . ABC acute.

From A draw AD I to BC or BC produced. Then must sq. on AC be less than the sum of sqq. on AB, BC, by twice rect. BC, BD.

For in Fig. 1 BC is divided into two parts in D, and in Fig. 2 BD is divided into two parts in C;

... in both cases sum of sqq. on BC, BD=sum of twice rect. BC, BD and

II. 7. Add to each the sq. on DA, then sum of sqq. on BC, BD, DA=sum of twice rect. BC, BD and sqq. on CD, DA;

.. sum of sqq. on BC, AB=sum of twice rect. BC, BD and sq. on AC;

I. 47. .. sq. on AC is less than sum of sqq. on AB, BC by twice rect. BC, BD.

The case, in which the perpendicular AD coincides with AC, needs no proof.

Q. E. D. Ex. Prove that the sum of the squares on any two sides of a triangle is equal to twice the sum of the squares on half the base and on the line joining the vertical angle with the middle point of the base,

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PROPOSITION XIV. PROBLEM.

To describe a square that shall be equal to a given rectilinear figure.

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Let A be the given rectil. figure.
It is reqd. to describe a square that shall=A.

Describe the rectangular O BCDE=A. I. 45.
Then if BENED the O BCDE is a square,

and what was reqd. is done. But if BE be not=ED, produce BE to F, so that EF=ED. Bisect BF in G; and with centre G and distance GB,

describe the semicircle BHF.

Produce DE to H and join GH. Then, : BF is divided equally in G and unequally in E, .. rect. BE, EF together with sq. on GE =sq. on GF

II. 5. =sq. on GH =sum of sqq. on EH, GE.

I. 47. Take from each the square on GE.

Then rect. BE, EF=sq. on EH.
But rect. BE, EF=BD, :: EF=ED;

:: sq. on EH=BD;
.. sq. on EH=rectil. figure A.

Q. E. F. Ex. Shew how to describe a rectangle equal to a given square, and having one of its sides equal to a given straight line.

Miscellaneous Exercises on Book II.

1. In a triangle, whose vertical angle is a right angle, a straight line is drawn from the vertex perpendicular to the base; shew that the square on either of the sides adjacent to the right angle is equal to the rectangle contained by the base and the segment of it adjacent to that side.

2. The squares on the diagonals of a parallelogram are together equal to the squares on the four sides.

3. If ABCD be any rectangle, and 0 any point either within or without the rectangle, shew that the sum of the squares on 04, OC is equal to the sum of the squares on OB, OD.

4. If either diagonal of a parallelogram be equal to one of the sides about the opposite angle of the figure, the square on it shall be less than the square on the other diameter, by twice the square on the other side about that opposite angle.

5. Produce a given straight line AB to C, so that the rectangle, contained by the sum and difference of AB and AC, may be equal to a given square.

6. Shew that the sum of the squares on the diagonals of any quadrilateral is less than the sum of the squares on the four sides, by four times the square on the line joining the middle points of the diagonals.

7. If the square on one perpendicular from the vertex of a triangle is equal to the rectangle, contained by the segments of the base, the vertical angle is a right angle.

8. Produce a given straight line so that the rectangle contained by the whole line thus produced and another given straight line may be equal to the square on the produced part.

9. ABC is a triangle right-angled at A ; in the hypotenuse two points D, E are taken such that BD=BA and CE=CA; shew that the square on DE is equal to twice the rectangle contained by BE, CD.

10. In any quadrilateral the squares on the diagonals are together equal to twice the sum of the squares on the straight lines joining the middle points of opposite sides.

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

12. CD is drawn perpendicular to AB, a side of the triangle ABC, in which AC=AB. Shew that the square on CD is equal to the square on BD together with twice the rectangle AD, DB.

13. If in any triangle BAC a line AD be drawn bisecting BC in D, shew that the sum of the squares on AB, AC is equal to twice the sum of the squares on AD, BD.

14. If ABC be an equilateral triangle, and AD, BE be perpendiculars to the opposite sides intersecting in F ; shew that the square on AB is equal to three times the square on AF.

15. Divide a given straight line into two parts, so that the rectangle contained by them shall be equal to the square described upon a straight line, which is less than half the line divided

NOTE 6.- On the Measurement of Areas.

To measure a Magnitude, we fix upon some magnitude of the same kind to serve as a standard or unit; and then any magnitude of that kind is measured by the number of times it contains this unit, and this number is called the MEASURE of the quantity.

Suppose, for instance, we wish to measure a straight line AB. We take another straight line EF for our standard,

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and then we say if AB contain EF three times, the measure of AB is 3, .four.......

.4, if Next suppose we wish to measure two straight lines AB, CD by the same standard EF. If

AB contain EF m times and

CD ... ..... n times, where m and n stand for numbers, whole or fractional, we say that AB and CD are commensurable.

But it may happen that we may be able to find a standard line EF, such that it is contained an exact number of times in AB; and yet there is no number, whole or fractional, which will express the number of times EF is contained in CD.

In such a case, where no unit-line can be found, such that it is contained an exact number of times in each of two lines AB, CD, these two lines are called incommensurable.

In the processes of Geometry we constantly meet with incommensurable magnitudes. Thus the side and diagonal of a square are incommensurables ; and so are the diameter and circumference of a circle.

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