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SYMMETRY.

Symmetry with respect to an axis.

173. Two points are said to be symmetrical with respect to a straight line, when the straight line bisects at right angles the straight line joining the two

points.

Thus, the two points P and P' are
symmetrical with respect to the line
MN, if MN bisects PP' at right angles. M
The straight line MN is called the

axis of symmetry.

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If we take the plane containing the pt. P, and turn it about the axis MN, until the upper part is brought down on the part below MN, the line AP will take the direction AP', and the point P will coincide with the point P'. Thus, when two points are symmetrical with respect to an axis, if one of the parts of the plane be revolved about the axis to bring it down on the other part, the symmetrical points coincide.

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174. Two figures are said to be symmetrical with respect to an axis, when every point in one figure has its symmetrical point in the other.

Thus, the figures ABC, A'B'C' are symmetrical with respect to the axis MN, if every point in the figure ABC has a symmetrical point in A'B'C' with respect to MN.

In all cases, two figures that are symmetrical with re

spect to an axis, can be applied one to the other, by revolving either about the axis; consequently they are equal.

The corresponding symmetrical lines of symmetrical figures are called homologous lines. Thus, in the symmetrical figures, ABC, A'B'C', the homologous lines are AB and A'B', BC and B'C', AC and A'C'.

Symmetry with respect to a point.

175. Two points are said to be symmetrical with respect to a third point, when this third point bisects the straight line joining the two points.

Thus, P and P' are symmetrical with

respect to A, if the straight line PP' is P A bisected at A.

The point A is called the centre of symmetry.

176. Two figures are said to be symmetrical with respect to a centre, when every point in one figure has its symmetrical point in the other.

Thus, the figures ABC, A'B'C' are symmetrical with respect to the centre O, if every point in the figure ABC has a symmetrical point in A'B'C'.

177. A figure is symmetrical with respect to an axis, when it can be divided by that axis into two figures symmetrical with respect to the axis.

A figure is symmetrical with respect to a centre, when every straight line drawn through that centre cuts the figure in two points symmetrical with respect to this centre.

A

B

Β'

Proposition 45. Theorem.

178. If a figure is symmetrical with respect to two axes at right angles to each other, it is also symmetrical with respect to their intersection as a centre.

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Hyp. Let the figure ABCDEFGH be symmetrical with respect to the two axes XX', YY', which intersect at O. To prove that O is the centre of symmetry of the figure. Proof. Let P be any pt. in the perimeter of the figure. Draw PRP' to XX', and PSQ to YY'.

Join RS, OP', and OQ.

Then, because the figure is symmetrical with respect to XX',

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Also, since the figure is symmetrical with respect to YY',

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.. SROQ is a

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;... RS is = and || to OQ.

Because both P'O and OQ are = and || to RS,

... the points P', O, Q are in the same st. line which is bisected at O.

.. any st. line P'OQ, drawn through O, is bisected at 0. .. the figure is symmetrical with respect to O as a centre (177).

Q.E, D.

EXERCISES.

The following theorems are given for the exercise of the student, that he may work out his own demonstrations. The student can make no solid acquisitions in geometry, without frequent practice in the application of the principles he has acquired. He should not merely learn the demonstration of a number of theorems, but he should acquire the power of grasping and demonstrating geometric theorems for himself, and this power can never be gained by memorizing demonstrations. He should understand that the ability to investigate, to reason for himself, is the chief object for the attainment of which he should strive. Diligent application, systematic practice in devising proofs of new propositions, is indispensable.

In the process of finding a demonstration the student should first construct a diagram, and state the hypothesis, including in the statement not only what the theorem says, but what it implies. He should also examine the conclusion, and see what it says and what it implies, and discover the relation between the hypothesis and the conclusion. A correct diagram is most useful in suggesting the steps by which a theorem is to be demonstrated. If the student will ask himself why he takes any particular step, he may avoid the habit of random guessing, and with more certainty discover the correct and direct process for effecting the demonstration. Sometimes it will be necessary to draw additional lines in the diagram, and to call to mind the different theorems which apply to the figure thus formed.

The demonstration must be framed in the simplest manner, but without omitting any logical step. This is a matter of practice, in which no general rule can be given.

The student should express each step of the demonstration completely and fully. The most common fault is that of passing over steps in the demonstration because the conclusion seems to be obvious,

It is often the case that the clearness of intuition acquired by a practised geometrician will make him impatient of the successive steps of detailed reasoning, and he will be eager to conduct his pupil to the desired end by a shorter and an easier road. But it is a great misfortune to the learner if he is deprived of that useful discipline in geometric reasoning which, however tedious it may seem in its application to short and easy propositions, is indispensable to the investigation of more lengthy and difficult ones.

One of the great objects of the study of geometry is to cultivate the habit of examining the logical foundations of those conclusions which are accepted without critical examination. The feeling of security that a conclusion is right before its foundation has been examined is a most fruitful source of erroneous opinions, and the person who neglects the habit of inquiring into what appears obvious is liable to pass over things which, had they been carefully examined, would have changed the conclusion.*

1. If the angles ABC and ACB at the base of an isosceles triangle be bisected by the lines BD, CD, show that DBC will be an isosceles triangle.

2. BAC is a triangle having the angle B double the angle A. If BD bisects the angle B and meets AC at D, show that BD is equal to AD.

3. In the triangle ABC, the angle A 50°, the angle B=70°. What angle will the bisectors of these two angles make with each other?

4. In the preceding triangle, what will be the values of the three exterior angles?

5. A given angle BAC is bisected; if CA is produced to G, and the angle BAG bisected, prove that the two bisecting lines are at right angles to each other.

6. ACB, ADB are two triangles on the same side of AB, such that AC BD, and AD BC, and AD and BC intersect at 0: prove that the triangle AOB is isosceles.

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7. ABC is a triangle, and the angle A is bisected by a line which meets BC at D: show that BA is greater than BD, and CA is greater than CD.

* Newcomb,

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