393. In a complete logical treatment every undefined term must occur in one or more axioms, since all knowledge of this term in a logical sense comes from the axioms in which it is found. In this text not every undefined term occurs in an axiom, for instance, the word between.

The axioms are, of course, based on our space intuition, or on our experience with the space in which we live. It is interesting to notice, however, that the axioms transcend that experience both as to exactness and extent. For instance, we have had no experience with endless lines, and hence we cannot know directly from experience whether or not there are complete lines which have no point in common. See §§ 89, 96.

A

394. Proofs are of two kinds, direct and indirect. direct proof starts with the hypothesis and leads step by step to the conclusion.

An indirect proof starts with the hypothesis and with the assumption that the conclusion does not hold, and shows that this leads to a contradiction with some known proposition. Or it starts simply with the assumption that the conclusion does not hold and shows that this leads to a contradiction with the hypothesis. This kind of proof is based upon the logical assumption that a proposition must either be true or not true. The proof consists in showing that if the proposition were not true, impossible consequences would follow. Hence the only remaining possibility is that it must be true.

395. Every proposition in geometry refers to some figure. See § 12. The essential characteristic of a figure is its description in words and not the drawing that represents it. Each drawing represents just one figure from a class of figures defined by the description. Thus we say

"let ABCD be a convex quadrilateral," and we construct a particular quadrilateral. We must then take care that all we say about it applies to any figure whatever so long as it is a convex quadrilateral. The logic of the proof must be entirely independent of the appearance of the constructed figure.

The description of the figure must contain all the conditions given by the hypothesis.

A good way to show that the description of the figure is what really enters into the proof, is to let one pupil describe the figure in words and each of the others draw a figure of his own to correspond to that description. The proof must then be such as to apply to every one of these figures though there may not be two of them exactly alike.

1. Every word in the language is defined in the dictionary. How is this possible in view of what has been said about the impossibility of defining every word?

2. Can we determine experimentally whether or not the space in which we live satisfies the parallel line axiom (§ 96)?

3. Can we determine experimentally whether or not there can be more than one straight line through two given points?

4. Which theorems of Chapter I are found by direct, proof and which by indirect proof?

5. If two triangles have two angles of the one equal to two angles of the other, and also any pair of corresponding sides equal, the triangles are congruent.

6. If two triangles have two sides of the one equal to two sides of the other, and also any pair of corresponding angles equal, the triangles are congruent in all cases except one. Discuss the various cases according as the given equal angles are greater than, equal to, or less than a right angle, and are, or are not, included between the equal sides, and thus discover the exceptional case.

7. State a theorem on the congruence of right triangles which is included in the preceding theorem.

8. What theorems of Chapter I on parallel lines can be proved without the parallel line axiom?

9. What theorems of Chapter I on parallelograms can be proved without the parallel line axiom?

10. What regular figures of the same kind can be used to exactly cover the plane about a point used as a vertex?

11. What combinations of the same or different regular figures can be used to exactly cover the plane about a point used as a vertex?

12. Suppose it has been proved that the base angles of an isosceles triangle are equal but that the converse has not been proved.

On this basis can it be decided whether or not the base angles are equal by simply measuring the sides?

Can it be decided on the same basis whether or not the sides are equal by simply measuring the base angles? Discuss fully.

13. The sum of the three medians of a triangle is less than the sum of the sides. See Ex. 34, p. 83.

14. The sum of the three altitudes of a triangle is less than the sum of the sides.

15. A triangle is isosceles (1) if an altitude and an angle-bisector coincide, (2) if an altitude and a median coincide, (3) if a median and an angle-bisector coincide.

16. If two sides of a triangle are unequal, the medians upon these sides are unequal and also the altitudes.

17. An isosceles triangle has two equal altitudes, D two equal medians, and two equal angle-bisectors.

18. ABCD is a square and the points E, F, G, H, are so taken that AE AH = CF CG. Prove that EFGH is a rectangle of constant perimeter, whatever the length of AE.

19. The bisectors of the exterior angles of a parallelogram form a rectangle the sum of the diagonals of which is the same as the sum of the sides of the parallelogram.

20. Prove that the perpendicular bisectors of the sides of a polygon inscribed in a circle meet in a point. Use this theorem to show that the statement is true of any triangle.

G с

21. The bisectors of the exterior angles of any quadrilateral form a quadrilateral whose opposite angles are supplementary.

Definition. In a polygon of n sides there are n angles and hence 2 n parts. Parts a, ZA, b, B, etc., are said to be consecutive if a lies on a side of A, b lies on the other side of A, and also on a side of ▲ B, etc.

F/f

a

E

e

B

C

22. Is the following proposition true? If in two polygons each of n sides 2 n-3 consecutive parts of one are equal respectively to 2 n-3 consecutive parts of the other, the polygons are congruent. SUGGESTION. Try to prove this proposition for n = 3, then for n = 4, and finally for the general polygon.

23. What theorems on the congruence of triangles are included in the preceding proposition?

A proposition may be proved not true by giving one example in which it does not hold.

24. Is the following proposition true? If in two polygons each of n sides 2 n 3 parts of one are equal respectively to 2 n - 3 corresponding parts of the other, the polygons are congruent provided at least one of the equal parts is a side.

25. Given three parallel lines, to construct an equilateral triangle whose vertices shall lie on these lines.

SOLUTION. Let P be any point on the middle line 2. Draw PC and PA, making an angle of 60° with 2. Through the points A, P, C construct a circle meeting l, in B. Then ABC is the required triangle.

SUGGESTION FOR PROOF. Compare BPA and BCA also & BPC and BAC.

26. Show how to modify the construction of the preceding example so as to make ABC similar to any given triangle.

27. If tangents are drawn to a circle at the extremities of a diameter and if another line tangent to the circle at P meets these two tangents in A and B respectively, show that AP PB = r2, where r is the radius of the circle.

LOCI CONSIDERATIONS.

397. Two methods are available to show that a certain geometric figure is the locus of points satisfying a given condition.

First method:

Prove (a) Every point satisfying the condition lies on the figure.

(b) Every point on the figure satisfies the condition.

Second method:

Prove (a') every point not on the figure fails to satisfy the condition. (b') Every point on the figure satisfies the condition.

The first of the methods is more direct and usually more simple. See § 127.

The second is likely to lead to proofs that are not general.

PROBLEMS AND APPLICATIONS.

B

1. AB is a fixed segment connecting two parallel lines and perpendicular to each of them. Find the locus of the vertices of all isosceles triangles whose common base is AB. Is the middle point of AB a part of this locus?

2. If in Ex. 1 AB is allowed to move always remaining perpendicular to the given lines, and if ABC is angle remaining fixed in shape, find the locus of the point C.

3. Find the locus of the centers of all parallelograms which have the same base and equal altitudes.

any triB' B

C

4. Find the locus of the centers of parallelograms obtained by cutting two parallel lines by parallel secant lines.

A

5. Find the locus of the vertices of all triangles which have the same base and equal areas.

6. Find the locus of a point whose distances from two intersecting lines are in a fixed ratio.

Note that the whole figure is symmetrical with respect to the bisector of the angle formed by the two given lines.