172. THEOREM. If a figure has two axes of symmetry at right angles to each other, their point of intersection is a center of symmetry of the figure.

Outline of Proof: It is to be shown that for every point P in the figure, a point p" also in the figure can be found such that PO=P"O and POP" is a

P"

Now use the hypothesis that ll', and each an axis of symmetry, to show that PP'p" p" is a rectangle of which DD" and D'D' are the diameters. Hence o, the intersection of DD" and D'D'", bisects the diagonal PP", making p" symmetric to P with respect to o, § 147, Ex. 2. Give the proof in full.

1. Has a square a center of symmetry? has a rectangle?

2. If a parallelogram has a center of symmetry, does it follow that it is a rectangle?

3. Has a trapezoid a center of symmetry? has an isosceles trapezoid?

4. If two non-parallel straight lines are symmetric with respect to a line 7, show that they meet this line in the same point and make equal angles with it. (Any point on the axis of symmetry is regarded as being symmetric to itself with respect to the axis.)

5. If segments AB and A'B' are symmetric with respect to a point O, they are equal and parallel.

6. Has a regular pentagon a center of symmetry? See figure of Ex. 6, § 167.

7. Has a regular hexagon a center of symmetry? Ex. 7, § 167. 8. Has an equilateral triangle a center of symmetry?

METHODS OF ATTACK.

174. No general rule can be given for proving theorems or for solving problems.

In the case of theorems the following suggestions may be helpful.

(1) Distinguish carefully the items of the hypothesis and of the conclusion.

It is best to tabulate these as suggested in § 79.

(2) Construct with care the figure described in the hypothesis.

The figure should be as general as the terms of the hypothesis permit. Thus if a triangle is called for but no special triangle is mentioned, then a scalene triangle should be drawn. Otherwise some particular form or appearance of the figure may lead to unwarranted conclusions.

(3) Study the hypothesis with care and determine whether any auxiliary lines may assist in deducing the properties required by the conclusion.

Study the theorems previously proved in this respect. A careful review of these proofs will lead to some insight as to how they were evolved.

175. Direct Proof. The majority of theorems are proved by passing directly from the hypothesis to the conclusion by a series of logical steps. This is called direct proof.

It is often helpful in discovering a direct proof to trace it backward from the conclusion.

Thus, we may observe that the conclusion C follows if statement B is true, and that B follows if A is true. If then we can show that A is true, it follows that B and C are true and the theorem is proved.

Having thus discovered a proof, we may then start from the beginning and follow it directly through.

As an example consider the following theorem:

The segments connecting the middle points of the opposite sides of any quadrilateral bisect

To prove that AB and CD bisect each other.

Ꭱ

D

Proof: Draw the diagonals PR and SQ and the segments AD, DB, BC, CA.

Now AB and CD bisect each other if ADBC is a □, and ADBC is aif AD | CB and AC || DB.

But AD CB since each is II SQ. See § 151.

Hence ADBC is a and AB and CD bisect each other.

Notice that the auxiliary lines PR and SQ divide the figure into triangles and this suggests the use of § 151.

176. Indirect Proof. In case a direct proof is not easily found, it is often possible to make a proof by assuming that the theorem is not true and showing that this leads to a conclusion known to be false.

As an example consider the follow

may have four acute angles, as 21, 22, 23, 24.

Extend the sides forming the exterior angles 5, 6, 7, 8.

37

Since 21+ 25 = 2 rt. 4, ≤2 + ≤6 = 2 rt. 4, etc., and since 21, 22, 23, 24 are all acute by hypothesis, it follows that 25, 26, 27, 28 are all obtuse, and hence 25 + 26 +27 + ≤8>4 rt. .

But this cannot be true since the sum of all the exterior angles is exactly 4 rt. 4 by § 162.

Hence the assumption that Z1, Z2, Z3, Z4 are all acute is false. That is, a convex polygon cannot have more than three acute angles.

The proof by the method of exclusion (§ 86) involves the indirect process in showing that all but one of the possible suppositions is false.

177. The solution of a problem often involves the same kind of analysis as that suggested for the discovery of a direct proof (§ 175).

For instance, consider the following problem:

To draw a line parallel to the base of a triangle such that the segment included between the sides shall equal the sum of the segments of the sides between the parallel and the base.

Given the ▲ ABC.

To find a point P through which to draw DE|| AB so that DP+ PE

= AD + EB.

Construction. Draw the bisectors of A and B meeting in point P.

Then P is the point required.

Proof: 123 since DE | AB, and 21 :

=

E

B

= 42 since BP bisects B.

This construction is discovered by observing that a point P must be found such that PE BE and PD = AD. This will be true if 23 22 and this follows if <1 = 22, while at the same time 26 = 25 and 24 = 25. Hence the bisectors of the base angles will determine the point P.

Having thus discovered the process, the construction and proof are made directly.

178. In general the most effective help is a ready knowledge of the facts of geometry already discovered, and skill in applying these will come with practice. It is important for this purpose that summaries like the following be made by the student and memorized:

179. (a) Two triangles are congruent if they have:

(1) Two sides and the included angle of the one equal to the corresponding parts of the other.

(2) Two angles and the included side of the one equal to the corresponding parts of the other.

(3) Three sides of the one equal respectively to three sides of the other.

In the case of right triangles :

(4) The hypotenuse and one side of one equal to the corresponding parts of the other.

(b) Two segments are proved equal if:

(1) They are homologous sides of congruent triangles. (2) They are legs of an isosceles triangle.

(3) They are opposite sides of a parallelogram. (4) They are radii of the same circle.

SUMMARY OF CHAPTER I.

1. Make a summary of ways in which two angles may be shown to be equal.

2. Make a summary of ways in which lines are proved parallel.

3. What conditions are sufficient to prove that a quadrilateral is a parallelogram?

4. Make a list of problems of construction thus far given.

5. Make a list of definitions thus far given. Which of the figures defined are found on page 4?

6. Tabulate all theorems on

(a) bisectors of angles and segments,

(b) perpendicular lines,

(c) polygons in general,

(d) symmetry.

7. What are some of the more important applications thus far given of the theorems in Chapter I?