The Teaching of Geometry Part 23

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If a carpenter's square is put on top of an upright stick, as here shown, and an observer sights along the arms to a distant point _B_ and a point _A_ near the stick, then the two triangles are similar. Hence _AD_ : _DC_ = _DC_ : _DB_. Hence, if _AD_ and _DC_ are measured, _DB_ can be found. The experiment is an interesting and instructive one for a cla.s.s, especially as the square can easily be made out of heavy pasteboard.

THEOREM. _If two chords intersect within a circle, the product of the segments of the one is equal to the product of the segments of the other._

THEOREM. _If from a point without a circle a secant and a tangent are drawn, the tangent is the mean proportional between the secant and its external segment._

COROLLARY. _If from a point without a circle a secant is drawn, the product of the secant and its external segment is constant in whatever direction the secant is drawn._

These two propositions and the corollary are all parts of one general proposition: _If through a point a line is drawn cutting a circle, the product of the segments of the line is constant_.

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If _P_ is within the circle, then _xx'_ = _yy'_; if _P_ is on the circle, then _x_ and _y_ become 0, and 0 _x'_ = 0 _y'_ = 0; if _P_ is at _P__{3}, then _x_ and _y_, having pa.s.sed through 0, may be considered negative if we wish, although the two negative signs would cancel out in the equation; if _P_ is at _P__{4}, then _y_ = _y'_ and we have _xx'_ = _y_^2, or _x_ : _y_ = _y_ : _x'_, as stated in the proposition.

We thus have an excellent example of the Principle of Continuity, and cla.s.ses are always interested to consider the result of letting _P_ a.s.sume various positions. Among the possible cases is the one of two tangents from an external point, and the one where _P_ is at the center of the circle.

Students should frequently be questioned as to the meaning of "product of lines." The Greeks always used "rectangle of lines," but it is entirely legitimate to speak of "product of lines," provided we define the expression consistently. Most writers do this, saying that by the product of lines is meant the product of their numerical values, a subject already discussed at the beginning of this chapter.

THEOREM. _The square on the bisector of an angle of a triangle is equal to the product of the sides of this angle diminished by the product of the segments made by the bisector upon the third side of the triangle._

This proposition enables us to compute the length of a bisector of a triangle if the lengths of the sides are known.

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For, in this figure, let _a_ = 3, _b_ = 5, and _c_ = 6.

Then [because] _x_ : _y_ = _b_ : _a_, and _y_ = 6 - _x_,

we have _x_/(6 - _x_) = 5/3.

[therefore] 3_x_ = 30 - 5_x_.

[therefore] _x_ = 3 3/4, _y_ = 2 1/4.

By the theorem, _z_^2 = _ab_ - _xy_ = 15 - (8 7/16) = 6 9/16.

[therefore] _z_ = [sqrt](6 9/16) = 1/4 [sqrt]105 = 2.5+.

THEOREM. _In any triangle the product of two sides is equal to the product of the diameter of the circ.u.mscribed circle by the alt.i.tude upon the third side._

This enables us, after the Pythagorean Theorem has been studied, to compute the length of the diameter of the circ.u.mscribed circle in terms of the three sides.

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For if we designate the sides by _a_, _b_, and _c_, as usual, and let _CD_ = _d_ and _PB_ = _x_, then

(_CP_)^2 = _a_^2 - _x_^2 = _b_^2 - (_c_ - _x_)^2.

[therefore] _a_^2 - _x_^2 = _b_^2 - _c_^2 + 2_cx_ - _x_^2.

[therefore] _x_ = (_a_^2 - _b_^2 + _c_^2) / 2_c_.

[therefore] (_CP_)^2 = _a_^2 - ((_a_^2 - _b_^2 + _c_^2) / 2_c_)^2.

But _CP_ _d_ = _ab_.

[therefore] _d_ = 2_abc_ / [sqrt](4_a_^2_c_^2 - (_a_^2 - _b_^2 + _c_^2)^2).

This is not available at this time, however, because the Pythagorean Theorem has not been proved.

These two propositions are merely special cases of the following general theorem, which may be given as an interesting exercise:

_If ABC is an inscribed triangle, and through C there are drawn two straight lines CD, meeting AB in D, and CP, meeting the circle in P, with angles ACD and PCB equal, then AC BC will equal CD CP._

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Fig. 1 is the general case where _D_ falls between _A_ and _B_.

If _CP_ is a diameter, it reduces to the second figure given on page 249. If _CP_ bisects [L]_ACB_, we have Fig. 3, from which may be proved the proposition given at the foot of page 248. If _D_ lies on _BA_ produced, we have Fig. 2. If _D_ lies on _AB_ produced, we have Fig. 4.

This general proposition is proved by showing that [triangles]_ADC_ and _PBC_ are similar, exactly as in the second proposition given on page 249.

These theorems are usually followed by problems of construction, of which only one has great interest, namely, _To divide a given line in extreme and mean ratio._

The purpose of this problem is to prepare for the construction of the regular decagon and pentagon. The division of a line in extreme and mean ratio is called "the golden section," and is probably "the section"

mentioned by Proclus when he says that Eudoxus "greatly added to the number of the theorems which Plato originated regarding the section."

The expression "golden section" is not old, however, and its origin is uncertain.

If a line _AB_ is divided in golden section at _P_, we have

_AB_ _PB_ = (_AP_)^2.

Therefore, if _AB_ = _a_, and _AP_ = _x_, we have

_a_(_a_ - _x_) = _x_^2, or _x_^2 + _ax_ - _a_^2 = 0; whence _x_ = - _a_/2 _a_/2[sqrt]5 = _a_(1.118 - 0.5) = 0.618_a_,

the other root representing the external point.

That is, _x_ = about 0.6_a_, and _a_ - _x_ = about 0.4_a_, and _a_ is therefore divided in about the ratio of 2 : 3.

There has been a great deal written upon the aesthetic features of the golden section. It is claimed that a line is most harmoniously divided when it is either bisected or divided in extreme and mean ratio. A painting has the strong feature in the center, or more often at a point about 0.4 of the distance from one side, that is, at the golden section of the width of the picture. It is said that in nature this same harmony is found, as in the division of the veins of such leaves as the ivy and fern.

FOOTNOTES:

[76] For a very full discussion of these four definitions see Heath's "Euclid," Vol. II, p. 116, and authorities there cited.