The Teaching of Geometry Part 21

THEOREM. _The bisector of an angle of a triangle divides the opposite side into segments which are proportional to the adjacent sides._

The proposition relating to the bisector of an exterior angle may be considered as a part of this one, but it is usually treated separately in order that the proof shall appear less involved, although the two are discussed together at this time. The proposition relating to the exterior angle was recognized by Pappus of Alexandria.

If _ABC_ is the given triangle, and _CP__{1}, _CP__{2} are respectively the internal and external bisectors, then _AB_ is divided harmonically by _P__{1} and _P__{2}.

[therefore]_AP__{1} : _P__{1}_B_ = _AP__{2} : _P__{2}_B_.

[therefore]_AP__{2} : _P__{2}_B_ = _AP__{2} - _P__{1}_P__{2} : _P__{1}_P__{2} - _P__{2}_B_,

and this is the criterion for the harmonic progression still seen in many algebras. For, letting _AP__{2} = _a_, _P__{1}_P__{2} = _b_, _P__{2}_B_ = _c_, we have

_a_/_c_ = (_a_ - _b_)/(_b_ - _c_),

which is also derived from taking the reciprocals of _a_, _b_, _c_, and placing them in an arithmetical progression, thus:

1/_b_ - 1/_a_ = 1/_c_ - 1/_b_,

whence (_a_ - _b_)/_ab_ = (_b_ - _c_)/_bc_,

or (_a_ - _b_)/(_b_ - _c_) = _ab_/_bc_ = _a_/_c_.

This is the reason why the line _AB_ is said to be divided harmonically. The line _P__{1}_P__{2} is also called the _harmonic mean_ between _AP__{2} and _P__{2}_B_, and the points _A_, _P__{1}, _B_, _P__{2} are said to form an _harmonic range_.

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It may be noted that [L]_P__{2}_CP__{1}, being made up of halves of two supplementary angles, is a right angle. Furthermore, if the ratio _CA_ : _CB_ is given, and _AB_ is given, then _P__{1} and _P__{2} are both fixed. Hence _C_ must lie on a semicircle with _P__{1}_P__{2} as a diameter, and therefore the locus of a point such that its distances from two given points are in a given ratio is a circle. This fact, Pappus tells us, was known to Apollonius.

At this point it is customary to define similar polygons as such as have their corresponding angles equal and their corresponding sides proportional. Aristotle gave substantially this definition, saying that such figures have "their sides proportional and their angles equal."

Euclid improved upon this by saying that they must "have their angles severally equal and the sides about the equal angles proportional." Our present phraseology seems clearer. Instead of "corresponding angles" we may say "h.o.m.ologous angles," but there seems to be no reason for using the less familiar word.

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It is more general to proceed by first considering similar figures instead of similar polygons, thus including the most obviously similar of all figures,--two circles; but such a procedure is felt to be too difficult by many teachers. By this plan we first define similar sets of points, _A__{1}, _A__{2}, _A__{3}, ..., and _B__{1}, _B__{2}, _B__{3}, ..., as such that _A__{1}_A__{2}, _B__{1}_B__{2}, _C__{1}_C__{2}, ...

are concurrent in _O_, and _A__{1}_O_ : _A__{2}_O_ = _B__{1}_O_ : _B__{2}_O_ = _C__{1}_O_ : _C__{2}_O_ = ... Here the constant ratio _A__{1}_O_ : _A__{2}_O_ is called the _ratio of similitude_, and _O_ is called the _center of similitude_. Having defined similar sets of points, we then define similar figures as those figures whose points form similar sets. Then the two circles, the four triangles, and the three quadrilaterals respectively are similar figures. If the ratio of similitude is 1, the similar figures become symmetric figures, and they are therefore congruent. All of the propositions relating to similar figures can be proved from this definition, but it is customary to use the Greek one instead.

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Among the interesting applications of similarity is the case of a shadow, as here shown, where the light is the center of similitude. It is also well known to most high school pupils that in a camera the lens reverses the image. The mathematical arrangement is here shown, the lens inclosing the center of similitude. The proposition may also be applied to the enlargement of maps and working drawings.

The propositions concerning similar figures have no particularly interesting history, nor do they present any difficulties that call for discussion. In schools where there is a little time for trigonometry, teachers sometimes find it helpful to begin such work at this time, since all of the trigonometric functions depend upon the properties of similar triangles, and a brief explanation of the simplest trigonometric functions may add a little interest to the work. In the present state of our curriculum we cannot do more than mention the matter as a topic of general interest in this connection.

It is a mistaken idea that geometry is a prerequisite to trigonometry.

We can get along very well in teaching trigonometry if we have three propositions: (1) the one about the sum of the angles of a triangle; (2) the Pythagorean Theorem; (3) the one that a.s.serts that two right triangles are similar if an acute angle of the one equals an acute angle of the other. For teachers who may care to make a little digression at this time, the following brief statement of a few of the facts of trigonometry may be of value:

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In the right triangle _OAB_ we shall let _AB_ = _y_, _OA_ = _x_, _OB_ = _r_, thus adopting the letters of higher mathematics. Then, so long as [L]_O_ remains the same, such ratios as _y_/_x_, _y_/_r_, etc., will remain the same, whatever is the size of the triangle. Some of these ratios have special names. For example, we call

_y_/_r_ the _sine_ of _O_, and we write sin _O_ = _y_/_r_;

_x_/_r_ the _cosine_ of _O_, and we write cos _O_ = _x_/_r_;

_y_/_x_ the _tangent_ of _O_, and we write tan _O_ = _y_/_x_.

Now because

sin _O_ = _y_/_r_, therefore _r_ sin _O_ = _y_;

and because cos _O_ = _x_/_r_, therefore _r_ cos _O_ = _x_;

and because tan _O_ = _y_/_x_, therefore _x_ tan _O_ = _y_.

Hence, if we knew the values of sin _O_, cos _O_, and tan _O_ for the various angles, we could find _x_, _y_, or _r_ if we knew any one of them.

Now the values of the sine, cosine, and tangent (_functions_ of the angles, as they are called) have been computed for the various angles, and some interest may be developed by obtaining them by actual measurement, using the protractor and squared paper. Some of those needed for such angles as a pupil in geometry is likely to use are as follows:

============================================================ ANGLE | SINE |COSINE|TANGENT|| ANGLE | SINE |COSINE|TANGENT ------+------+------+-------++-------+------+------+-------- 5 | .087 | .996 | .087 || 50 | .766 | .643 | 1.192 ------+------+------+-------++-------+------+------+-------- 10 | .174 | .985 | .176 || 55 | .819 | .574 | 1.428 ------+------+------+-------++-------+------+------+-------- 15 | .259 | .966 | .268 || 60 | .866 | .500 | 1.732 ------+------+------+-------++-------+------+------+-------- 20 | .342 | .940 | .364 || 65 | .906 | .423 | 2.145 ------+------+------+-------++-------+------+------+-------- 25 | .423 | .906 | .466 || 70 | .940 | .342 | 2.748 ------+------+------+-------++-------+------+------+-------- 30 | .500 | .866 | .577 || 75 | .966 | .259 | 3.732 ------+------+------+-------++-------+------+------+-------- 35 | .574 | .819 | .700 || 80 | .985 | .174 | 5.671 ------+------+------+-------++-------+------+------+-------- 40 | .643 | .766 | .839 || 85 | .996 | .087 |11.430 ------+------+------+-------++-------+------+------+-------- 45 | .707 | .707 | 1.000 || 90 | 1.00 | .000 |[infinity]

It will of course be understood that the values are correct only to the nearest thousandth. Thus the cosine of 5 is 0.99619, and the sine of 85 is 0.99619. The entire table can be copied by a cla.s.s in five minutes if a teacher wishes to introduce this phase of the work, and the author has frequently a.s.signed the computing of a simpler table as a cla.s.s exercise.

Referring to the figure, if we know that _r_ = 30 and [L]_O_ = 40, then since _y_ = _r_ sin _O_, we have _y_ = 30 0.643 = 19.29. If we know that _x_ = 60 and [L]_O_ = 35, then since _y_ = _x_ tan _O_, we have _y_ = 60 0.7 = 42. We may also find _r_, for cos _O_ = _x_/_r_, whence _r_ = _x_/(cos _O_) = 60/0.819 = 73.26.

Therefore, if we could easily measure [L]_O_ and could measure the distance _x_, we could find the height of a building _y_. In trigonometry we use a transit for measuring angles, but it is easy to measure them with sufficient accuracy for ill.u.s.trative purposes by placing an ordinary paper protractor upon something level, so that the center comes at the edge, and then sighting along a ruler held against it, so as to find the angle of elevation of a building. We may then measure the distance to the building and apply the formula _y_ = _x_ tan _O_.

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Finaeus's "De re et praxi geometrica," Paris, 1556]

It should always be understood that expensive apparatus is not necessary for such ill.u.s.trative work. The telescope used on the transit is only three hundred years old, and the world got along very well with its trigonometry before that was invented. So a little ingenuity will enable any one to make from cheap protractors about as satisfactory instruments as the world used before 1600. In order that this may be the more fully appreciated, a few ill.u.s.trations are here given, showing the old instruments and methods used in practical surveying before the eighteenth century.

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The ill.u.s.tration on page 236 shows a simple form of the quadrant, an instrument easily made by a pupil who may be interested in outdoor work. It was the common surveying instrument of the early days. A more elaborate example is seen in the ill.u.s.tration, on page 237, of a seventeenth-century bra.s.s specimen in the author's collection.[77]

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Bartoli's "Del modo di misurare," Venice, 1689]

Another type, easily made by pupils, is shown in the above ill.u.s.tration from Bartoli, 1689. Such instruments were usually made of wood, bra.s.s, or ivory.[78]

Instruments for the running of lines perpendicular to other lines were formerly common, and are easily made. They suffice, as the following ill.u.s.tration shows, for surveying an ordinary field.

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