NOTE 15. Another Proof of I. 24.

In the As ABC, DEF, let AB=DE_and_AC=DF, and let BAC be greater than EDF. Then must BC be greater than EF.

Apply the ▲ DEF to the ▲ ABC
so that DE coincides with AB.
Then . EDF is less than 4 BAC,
DF will fall between BA and AC,
and F will fall on, or above, or below, BC.
I. If F fall on BC,

Then BO, OF together are greater than BF,

and OC, AO..

.AC;

I. 20.

I. 20.

.BF, AC together,

.. BC is greater than BF;

and .. EF is less than BC.

Q. E. D.

S.E.

NOTE 16. Euclid's Proof of I. 26.

If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, viz., either the sides adjacent to the equal angles, or the sides opposite to equal angles in each; then shall the other sides be equal, each to each; and also the third angle of the one to the third angle of the other.

Let 4 ABC = DEF, and 4 ACB = LDFE;

and first,

Let the sides adjacent to the equals in each be equal, that is, let BC=EF.

Then must AB=DE, and AC=DF, and ▲ BAC = LEDF.

For if AB be not = DE, one of them must be the greater. Let AB be the greater, and make GB=DE, and join GC. Then in As GBC, DEF,

GB=DE, and BC=EF, and 4 GBC:

But

.. 4 GCB=LDFE.

= L DEF,

I. 4.

LACB= L DFE by hypothesis;

.. 4 GCB= L ACB;

that is, the less-the greater, which is impossible.

.. AB is not greater than DE.

In the same way it may be shewn that AB is not less than

DE;

.. AB=DE.

Then in As ABC, DEF,

· AB=DE, and BC= EF, and 2 ABC= 2 DEF,

.. AC=DF, and ▲ BAC=1 EDF.

I. 4.

Next, let the sides which are opposite to equal angles in each triangle be equal, viz., AB=DE.

Then must AC=DF, and BC=EF, and ▲ BAC =

= LEDF.

For if BC be not=EF, let BC be the greater, and make BH=EF, and join AH.

Then in AS ABH, DEF,

· AB=DE, and BH-EF, and ▲ ABH= ▲ DEF,

.. LAHB= L DFE.

But ACB = LDFE, by hypothesis,

.. LAHB: = L ACB;

I. 4.

that is, the exterior ▲ of ▲ AHC is equal to the interior and opposite ACB, which is impossible.

.. BC is not greater than EF.

In the same way it may be shewn that BC is not less than

· AB=DE, and BC=EF, and ▲ ABC= ▲ DEF,

.. AC=DF, and ▲ BAC

EDF.

I. 4.

Q. E. D.

Miscellaneous Exercises on Books I. and II.

1. AB and CD are equal straight lines, bisecting one another at right angles. Shew that ACBD is a square.

2. From a point in the side of a parallelogram draw a line dividing the parallelogram into two equal parts.

3. Draw through a point, lying between two lines that intersect, a line terminated by the given lines, and bisected in the given point.

4. The square on the hypotenuse of an isosceles right-angled triangle is equal to four times the square on the perpendicular from the right angle on the hypotenuse.

5. Describe a rhombus, which shall be equal to a given triangle, and have each of its sides equal to one side of the triangle.

6. Shew how to describe a square, when the difference between the lengths of a diagonal and a side is given.

7. Two rings slide on two straight lines, which intersect at right angles in a point O, and are connected by an inextensible string passing round a peg fixed at that point. Shew that the rings will be nearest to each other when they are equidistant from 0.

8. ABCD is a parallelogram, whose diagonals AC, BD intersect in O; shew that if the parallelograms AOBP, DOCQ be completed, the straight line joining P and Q passes through 0.

9. ABCD, EBCF are two parallelograms on the same base BC, and so situated that CF passes through A. Join DF, and produce it to meet BE produced in K; join FB, and prove that the triangle FAB equals the triangle FEK.

10. The alternate sides of a polygon are produced to meet; shew that all the angles at their points of intersection together with four right angles are equal to all the interior angles of the polygon.

11. Shew that the perimeter of a rectangle is always greater than that of the square equal to the rectangle.

12. Shew that the opposite sides of an equiangular hexagon are parallel, though they be not equal; and that any two sides that are adjacent are together equal to the two which are parallel.

13. If two equal straight lines intersect each other anywhere at right angles, shew that the area of the quadrilateral formed by joining their extremities is invariable, and equal to one-half the square on either line.

14. Two triangles ACB, ADB are constructed on the same side of the same base AB. Shew that if AC-BD and AD=BC, then CD is parallel to AB; but if AC=BC and AD=BD, then CD is perpendicular to AB.

15. AB is the hypotenuse of a right-angled triangle ABC: find a point D in AB, such that DB may be equal to the perpendicular from D on AC.

16. Find the locus of the vertices of triangles of equal area on the same base.

17. Shew that the perimeter of an isosceles triangle is less than that of any triangle of equal area on the same base.

18. If each of the equal angles of an isosceles triangle be equal to one-fourth the vertical angle, and from one of them a perpendicular be drawn to the base, meeting the opposite side produced, then will the part produced, the perpendicular, and the remaining side, form an equilateral triangle.

19. If a straight line terminated by the sides of a triangle be bisected, shew that no other line terminated by the same two sides can be bisected in the same point.

20. From a given point draw to two parallel straight lines two equal straight lines at right angles to each other.

21. Given the lengths of the two diagonals of a rhombus, construct it.

22. ABCD is a quadrilateral figure: construct a triangle whose base shall be in the line AB, such that its altitude shall be equal to a given line, and its area equal to that of the quadrilateral.

23. If ABC be a triangle in which C is a right angle, shew how to draw a straight line parallel to a given straight line, so as to be terminated by CA and CB and bisected by AB.