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Proposition 25.- This proposition is the converse of Prop. 24. In Prop. 24 we take for granted that the angle BAC is greater than the angle EDF, and we prove that the base BC is greater than the base EF; conversely in Prop. 25 we take for granted the base and prove the angle. Proposition 25 should be compared with Prop. 8. In Prop. 8 the angle BAC is proved equal to the angle EDF, and in Prop. 25 the angle BAC is proved greater than the angle EDF. Prop. 25 is proved like Prop. 19 the method of exhaustion. The two impossible cases depend on Props. 4 and 24.
Proposition 26.—This really contains two propositions, which we give as Case I. and Case II. Both are proved by a Reductio ad Absurdum, in Case I. the Ninth Axiom, and in Case II. Prop. 16 being used. The Proofs of both cases are made up of two simple applications of Prop. 4.
It might be supposed that a Case III. should be mentioned, namely, when the side AC is given equal to the side DF. But this is really the same as Case II., the equal sides being opposite to equal angles.
EASY EXERCISES ON PROPOSITIONS 21—26. 1. Take any point 0 within the triangle ABC and join OA, OB, OC. Prore that OA, OB, OC are together less than AB, BC, CA.
2. Describe an angle double of a given angle.
3. Show how to construct a quadrilateral figure equal in all respects to a given quadrilateral figure.
4. In the fig. of Prop. 21 join AD and produce it to meet the base BC in F; by this means prove that the angle BDC is greater than the angle BAC.
5. Take any finite straight line AB and any two rectilineal angles C and D; show how to construct on AB a triangle having two of its angles equal to C and D. When is this impossible ?
6. In the fig. of Prop. 24 bisect the angle FDG by the straight line DH, which meets EG in H, and join FH. Prove that FH is equal to HG, and hence prove that BC is greater than EF.
7. In the triangle ABC join the vertex A to M the middle point of the base BC, and prove that the angle AMB is greater than, equal to, or less than the angle AMC according as AB is greater than, equal to, or less than AC.
8. ABC is an isosceles triangle having the side AB equal to the side AC. Bisect each of the angles ABC, ACB by the straight lines BY, CZ which meet AC and AB in Y and Z respectively. Prove that the triangles YBO ZCB are equal in all respects.
9. Prove by Reductio ad Absurdum, as in Prop. 26, that if two right-angled triangles have their hypotenuses equal and one side equal to one side, the triangles are equal in all respects.
SUMMARY OF THE RESULTS ARRIVED AT IN EUCLID I. 1—26.
The first 26 Propositions in Book I. deal with Lines, Angles, and Triangles. Lines.-(a) Props. 2, 3, 10, 11, 12 are Problems on the drawing
and bisecting of lines. (6) Props. 13, 14, 15 are Theorems proving that if
two lines cut each other the angles on one side are together equal to two right angles, and that
the vertically opposite angles are equal. Angles.- Props. 9 and 23 are Problems on bisecting or describing
an angle. Triangles.-(a) Props. 1 and 22 are Problems on describing a
triangle. (6) Props. 5 and 6 are Theorems, dealing with an
isosceles triangle. (c) Props. 16, 17, 18, 19, 20, 21 are Theorems about
a single triangle, and the chief results arrived
at are that(1) The exterior angle is greater than either interior
opposite angle. (2) Any two angles are less than two right angles. (3) The greater side and greater angle are opposite each
other. (4) Any two sides of a triangle are greater than the third.
(d) Props. 4, 8, 26 are Theorems dealing with the
equality of two triangles; and 24, 25 are Theorems dealing with the inequality of two
triangles. Of all the Propositions in this part of Euclid, Props. 4, 8, 26 are the most important. In order to prove two triangles equal in all respects, we must have given at least three parts of one triangle equal to three parts of the other; but even then in two cases the triangles may not be equal in all respects, as the following diagrams will show.
CASE VI.—Two sides and an angle not included by them equal
Triangles are not necessarily
equal, for there may be two
In order therefore to prove two triangles equal in all respects recourse must be had to one of the first four of these Cases,
ANALYSIS OF THE METHODS OF PROOF
EMPLOYED IN EUCLID I. 1—26.
The Methods of Proof employed in Book I., Props. 1—26, may be arranged in five classes as follows :
I. Axiom 1.--Props. 1, 2, 3, 13, and 22 all employ Axiom I.
II. Superposition.—Props. 4 and 8 are proved by supposing one triangle placed on the other.
III. “Much more then.”—Props. 7 and 24 are proved by what may be called the Much more then” method of proof.
IV. Exhaustion.—Props. 19 and 25 are proved by the method of exhaustion, that is, by showing that every case is impossible except one.
V. Riders. The proofs of Props. 5, 6, 9, 10, 11, 12, 14, 15, 16, 17, 18, 20, 21, 23, 24, and 26 all depend more or less on some previous proposition, and they may be grouped as follows :
(a) Reductio ad Absurdum.—Props. 6, 14, and 26 employ the method of Reductio ad Absurdum.
(6) Proposition 4.- Props. 5, 6, 10, 16, 24, and 26 depend on Prop. 4.
A more detailed account of the Method of Proof employed in each Proposition will be found in the Notes to the various Propositions.
ON THE SOLUTION OF RIDERS. It is quite a mistake to suppose that Geometry consists of two distinct parts, viz. Propositions which are proved in Euclid, and Riders which are less important and are proved by means of the Propositions given in Euclid. As a matter of fact, there are Propositions in Euclid, such as I. 7, which are of very little importance ; and there are so-called Riders not proved in Euclid which are of the greatest importance. Moreover, the Analysis of the Methods of Proof employed in Euclid I. 1—26 (see page 54) has shown that the majority of the Propositions in Euclid are really Riders.
The Propositions therefore should rather be studied as examples of Methods of Proof, and with the idea that the same methods may be used in other cases.
I. In attempting to solve Riders, the student often finds a difficulty in the general terms in which Riders are usually given. To overcome this difficulty, the learner should accustom himself to draw figures. It is a good plan to draw the figures of all the Propositions in Euclid, and to draw them fairly large, say at least six inches in height, so that any glaring defect in the drawing will evident at once. Figures should also be drawn for all the definitions.
If this has been done, little difficulty will be found in drawing figures for Riders, and the first step in solving a Rider is to draw a particular figure with letters to mark particular angles, lines, etc.
EXERCISES. Draw, without proof, complete figures with letters for the following Riders :
1. The straight line, which joins the middle point of the base of an isosceles triangle to the vertex, bisects the vertical angle.
2. The straight line which joins the centres of two intersecting circles is perpendicular to the common chord of the circles.
3. The straight line which joins the vertices of isosceles triangles on the same base and on opposite sides of it, is perpendicular to the base.
4. The diagonals of a rhombus bisect each other at right angles. 5. The three medians of a triangle pass through the same point.
6. The perpendicular is the shortest line that can be drawn from a given point to a given straight line.
II. After drawing the figure for the Rider the student should carefully note down (a) everything which may be taken for granted, and (6) what is required to be done or proved. And in doing this, care should be taken that the figure does not assume more than is allowed, e.y. an equilateral triangle must not be drawn for an isosceles triangle ; two circles which intersect must not be drawn both of the same size unless this equality is expressly mentioned.
EXERCISE. In the above Exercises, state in each case what is taken for granted and what has to be proved.