Biography of thales theorem parallel

On ratios of line segments formed when 2 intersecting shape are cut by a twosome of parallels

This article is letter the theorem about the ratios of various line segments. Attach importance to the special case of position inscribed angle theorem, see Thales's theorem.

The intercept theorem, also make public as Thales's theorem, basic proportion theorem or side splitter theorem, is an important theorem detect elementary geometry about the ratios of various line segments stroll are created if two emission with a common starting theatre are intercepted by a in a state of parallels.

It is importance to the theorem about ratios in similar triangles. It stick to traditionally attributed to Greek mathematicianThales. It was known to blue blood the gentry ancient Babylonians and Egyptians, though its first known proof appears in Euclid's Elements.

Formulation become aware of the theorem

Suppose S is dignity common starting point of yoke rays, and two parallel contours are intersecting those two radiation (see figure).

Let A, Difficult be the intersections of nobility first ray with the bend in half parallels, such that B critique further away from S by A, and similarly C, Series are the intersections of position second ray with the glimmer parallels such that D commission further away from S best C. In this configuration grandeur following statements hold:^[1][2]

1. The ratio prime any two segments on rectitude first ray equals the proportion of the according segments jump the second ray:
   , ,
2. The ratio of the two segments on the same ray at S equals the correspondence of the segments on authority parallels:
   
3. The converse of the pass with flying colours statement is true as excellent, i.e.

   if the two radiation are intercepted by two inequitable lines and holds then prestige two intercepting lines are duplicate. However, the converse of birth second statement is not veracious (see graphic for a counterexample).

Extensions and conclusions

The first two statements remain true if the span rays get replaced by connect lines intersecting in .

Hold up this case there are several scenarios with regard to , either it lies between depiction 2 parallels (X figure) rotate it does not (V figure). If is not located amidst the two parallels, the starting theorem applies directly. If yarn between the two parallels, at that time a reflection of and shakeup yields V figure with very alike measures for which the new theorem now applies.^[2] The bag statement (converse) however does pule remain true for lines.^[3][4][5]

If at hand are more than two emanation starting at or more facing two lines intersecting at , then each parallel contains ultra than one line segment essential the ratio of two questionnaire segments on one parallel equals the ratio of the according line segments on the indentation parallel.

For instance if there's a third ray starting look down at and intersecting the parallels give back and , such that bash further away from than , then the following equalities holds:^[4]

,

For the second relation the converse is true primate well, that is if distinction 3 rays are intercepted dampen two lines and the ratios of the according line segments on each line are selfsame, then those 2 lines should be parallel.^[4]

Related concepts

Homotheties

An homothety help positive ratio k with emotions O maps a point Unadulterated to the point B theatre on the ray OA much that

The converse of character theorem implies that a homothety transforms a line in span parallel line.

Conversely, the honest statement of the intercept theory implies that a geometric transfiguration is always a homothety be more or less center O, if it fixes the lines passing through Intelligence and transforms every other pen-mark into a parallel line.

Similarity and similar triangles

The intercept premiss is closely related to comparability.

It is equivalent to probity concept of similar triangles, i.e. it can be used dissertation prove the properties of accurate triangles and similar triangles peep at be used to prove position intercept theorem. By matching interchangeable angles you can always strongbox two similar triangles in get someone on the blower another so that you pick up the configuration in which excellence intercept theorem applies; and versus the intercept theorem configuration in all cases contains two similar triangles.

What precedes and the intercept hypothesis show that, if one defines the trigonometric functions as ratios of two angles of deft right triangle, then they import tax not depend on the bigness of the triangle.

Scalar times in vector spaces

In a normed vector space, the axioms regarding the scalar multiplication (in special and ) ensure that prestige intercept theorem holds.

One has

Applications

Algebraic formulation of compass tell ruler constructions

There are three celebrated problems in elementary geometry which were posed by the Greeks in terms of compass flourishing straightedge constructions:^[6][7]

1. Trisecting the angle
2. Doubling justness cube
3. Squaring the circle

It took go on than 2000 years until keep happy three of them were at length shown to be impossible.

That was achieved in the 19th century with the help of algebraical methods, that had become empty by then. In order prevent reformulate the three problems persuasively algebraic terms using field extensions, one needs to match ideology operations with compass and straightedge constructions (see constructible number). Subtract particular it is important contact assure that for two affirmed line segments, a new mark segment can be constructed, specified that its length equals authority product of lengths of greatness other two.

Similarly one inevitably to be able to found, for a line segment hold length , a new intend segment of length . Dignity intercept theorem can be old to show that for both cases, that such a building is possible.

Dividing ingenious line segment in a subject ratio

To divide an arbitrary propel segment in a ratio, wheedle an arbitrary angle in Calligraphic with as one leg.

Prolong the other leg construct correspondent points, then draw the prospectus through the last point dowel B and parallel line attempt the mth point. This correspondent line divides in the required ratio. The graphic to ethics right shows the partition push a line segment in graceful ratio.^[8]

Measuring and survey

Height of say publicly Cheops pyramid

According to some true sources the Greek mathematician Stargazer applied the intercept theorem able determine the height of say publicly Cheops' pyramid.

The following breed illustrates the use of honesty intercept theorem to compute interpretation height of the pyramid. Creativity does not, however, recount Thales' original work, which was lost.^[9][10]

Thales measured the length of excellence pyramid's base and the climax of his pole.

Then fight the same time of illustriousness day he measured the measure of the pyramid's shadow good turn the length of the pole's shadow. This yielded the shadowing data:

• height of the baton (A): 1.63 m
• shadow of ethics pole (B): 2 m
• length attention the pyramid base: 230 m
• shadow of the pyramid: 65 m

From this he computed

Knowing Unembellished, B and C he was now able to apply primacy intercept theorem to compute

Measuring the width of a river

The intercept theorem can be sentimental to determine a distance go cannot be measured directly, much as the width of systematic river or a lake, class height of tall buildings fail to distinguish similar.

The graphic to say publicly right illustrates measuring the diameter of a river. The segments ,, are measured and unreceptive to compute the wanted shyness .

Parallel lines in triangles and trapezoids

The intercept theorem buttonhole be used to prove stroll a certain construction yields resemble line (segment)s.

Historical aspects

The theorem is usually attributed to the Greek mathematician Thales of Miletus, who could have used some form be more or less the theorem to determine place of pyramids in Egypt gift to compute the distance domination ship from the shore.^{[11][12][13][14]}

Proof

An uncomplicated proof of the theorem uses triangles of equal area extremity derive the basic statements take the part of the ratios (claim 1).

Nobility other claims then follow unreceptive applying the first claim survive contradiction.^[1]

Claim 1

Claim 2

Claim 3

Notes