PROPORTION\@. SIMILAR POLYGONS

Excerpts

• PROPORTION\@. SIMILAR POLYGONS

  A **proportion** is an expression of equality between two equal ratios; and is written in one of the following forms:

• PROPORTION\@. SIMILAR POLYGONS

  In every proportion the product of the extremes is equal to the product of the means.

• PROPORTION\@. SIMILAR POLYGONS

  If three quantities are in continued proportion, the second is called the **mean proportional** between the other two, and the third is called the **third proportional** to the other two.

• PROPORTION\@. SIMILAR POLYGONS

  The fourth proportional to three given quantities is the fourth term of the proportion which has for its first three terms the three given quantities *taken in order.*

• PROPORTION\@. SIMILAR POLYGONS

  If the product of two quantities is equal to the product of two others, either two may be made the extremes of the proportion in which the other two are made the means.

• PROPORTION\@. SIMILAR POLYGONS

  If four quantities are in proportion, they are in proportion by **inversion**; that is, the second term is to the first as the fourth is to the third.

• PROPORTION\@. SIMILAR POLYGONS

  Thus, in the proportion $a:b = b:c$; $b$ is the mean proportional between $a$ and $c$; and $c$ is the third proportional to $a$ and $b$.

• PROPORTION\@. SIMILAR POLYGONS

  The mean proportional between two quantities is equal to the square root of their product.

• PROPORTION\@. SIMILAR POLYGONS

  In the treatment of proportion, it is assumed that the *quantities* involved are expressed by their *numerical measures*.

• PROPORTION\@. SIMILAR POLYGONS

  By increasing the *number* of equal parts into which $AE$ is divided, we can make the *length* of each part less than any assigned value, however small, but not zero.

• PROPORTION\@. SIMILAR POLYGONS

  **Similar polygons** are polygons that have their homologous angles equal, and their homologous sides proportional.

• PROPORTION\@. SIMILAR POLYGONS

  The primary idea of similarity is **likeness of form**.

• PROPORTION\@. SIMILAR POLYGONS

  The sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse.

• PROPORTION\@. SIMILAR POLYGONS

  A straight line is divided **in extreme and mean ratio**, when one of the segments is the mean proportional between the whole line and the other segment.

• PROPORTION\@. SIMILAR POLYGONS

  If from a fixed 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.

• PROPORTION\@. SIMILAR POLYGONS

  The perpendicular is the mean proportional between the segments of the hypotenuse.

• PROPORTION\@. SIMILAR POLYGONS

  The last three theorems enable us to compute the lengths of the altitudes of a triangle if the lengths of the three sides are known.

• PROPORTION\@. SIMILAR POLYGONS

  The **projection** of any line upon a second line is the segment of the second line included between the perpendiculars drawn to it from the extremities of the first line.

• PROPORTION\@. SIMILAR POLYGONS

  that is, the ratio of two corresponding segments is equal to the *reciprocal* of the ratio of the other two segments.

Equations

• PROPORTION\@. SIMILAR POLYGONS

  a:b = c:d

  Four quantities form a proportion when the ratio of the first to the second equals the ratio of the third to the fourth.

• PROPORTION\@. SIMILAR POLYGONS

  a:b = b:c = c:d = d:e

  Quantities a, b, c, d, e are in continued proportion when each consecutive pair has the same ratio.

• PROPORTION\@. SIMILAR POLYGONS

  a+c+e+g : b+d+f+h = a:b

  In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent.

• PROPORTION\@. SIMILAR POLYGONS

  \dfrac{EB}{AE} = \dfrac{FC}{AF}

  In the incommensurable case, by a limiting argument, the ratio EB to AE equals the ratio FC to AF.

• PROPORTION\@. SIMILAR POLYGONS

  \dfrac{AB}{A'B'}

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• PROPORTION\@. SIMILAR POLYGONS

  AB:A'B' = BC:B'C' = CD:C'D'

  Similar polygons have their homologous angles equal and their homologous sides proportional, so the ratios of corresponding sides are equal.

• PROPORTION\@. SIMILAR POLYGONS

  AB:A'B' = AC:A'C' = BC:B'C'

  Two mutually equiangular triangles are similar, so their corresponding sides are proportional.

• PROPORTION\@. SIMILAR POLYGONS

  \dfrac {CO}{C'O'}=\dfrac {AC}{A'C'}=\dfrac {AB}{A'B'}=\dfrac {BC}{B'C'}

  The homologous altitudes of two similar triangles have the same ratio as any two homologous sides.

• PROPORTION\@. SIMILAR POLYGONS

  \dfrac{AB}{A'B'}= \dfrac{BC}{B'C'}= \dfrac{CD}{C'D'}= \dfrac{DE}{D'E'}

  Two parallels cut by three or more transversals through one point have proportional corresponding segments.

• PROPORTION\@. SIMILAR POLYGONS

  \overline{AC}^2=AB × AF

  In a right triangle, the square of a leg equals the hypotenuse times the segment of the hypotenuse adjacent to that leg.

• PROPORTION\@. SIMILAR POLYGONS

  \dfrac{\overline{AC}^2}{\overline{BC}^2} = \dfrac{AB × AF}{AB × BF} = \dfrac{AF}{BF}

  The squares of the two legs of a right triangle are proportional to the segments of the hypotenuse adjacent to them.

• PROPORTION\@. SIMILAR POLYGONS

  \dfrac{\overline{AB}^2}{\overline{AC}^2} = \dfrac{AB × AB}{AB × AF} = \dfrac{AB}{AF}

  The square of the hypotenuse over the square of a leg equals the hypotenuse over the segment of the hypotenuse adjacent to that leg.

• PROPORTION\@. SIMILAR POLYGONS

  AB(AF + BF) = \overline{AB}^2

  The squares of the two legs of a right triangle sum to the square of the hypotenuse; this is the closing algebraic step of the proof (the sum of squares is written across two braces in the source).

• PROPORTION\@. SIMILAR POLYGONS

  \overline{AC}^2 = \overline{AB}^2 + \overline{BC}^2 = 2 \overline{AB}^2

  For a square ABCD, the square of the diagonal AC equals twice the square of a side AB.

• PROPORTION\@. SIMILAR POLYGONS

  AC = AB \sqrt{2}

  The diagonal of a square is the side times the square root of two, so diagonal and side are incommensurable.

• PROPORTION\@. SIMILAR POLYGONS

  \overline{AB}^2 = \overline{BC}^2 + \overline{AC}^2 - 2 BC × DC

  In any triangle, the square of the side opposite an acute angle C equals the sum of the squares of the other two sides minus twice the product of one of them and the projection of the other upon it.

• PROPORTION\@. SIMILAR POLYGONS

  \overline{AB}^2 = \overline{BC}^2 + \overline{AC}^2 + 2 BC × DC

  In an obtuse triangle, the square of the side opposite the obtuse angle C equals the sum of the squares of the other two sides plus twice the product of one of them and the projection of the other upon it.

• PROPORTION\@. SIMILAR POLYGONS

  a^2+b^2 = 2m^2+2\left(\dfrac{c}{2}\right)^2

  In a triangle with sides a, b, c and median m drawn to side c, the sum of the squares of a and b equals twice the square of half of c plus twice the square of the median.

• PROPORTION\@. SIMILAR POLYGONS

  \dfrac{OM}{OQ} = \dfrac{OP}{ON}

  When two chords intersect in a circle, the ratio of two corresponding segments equals the reciprocal of the ratio of the other two; the segments are reciprocally proportional.

• PROPORTION\@. SIMILAR POLYGONS

  AC : AD = AD : AB

  A tangent from an external point is the mean proportional between the whole secant and its external segment.

• PROPORTION\@. SIMILAR POLYGONS

  AC × AB = \overline{AD}^2

  From a fixed point outside a circle, the product of a secant and its external segment equals the square of the tangent, so it is constant for every secant.

• PROPORTION\@. SIMILAR POLYGONS

  \overline{NO}^2 = NM × NP - OM × OP

  The square of the bisector of an angle of a triangle equals the product of the two sides enclosing the angle minus the product of the segments the bisector makes on the third side.

• PROPORTION\@. SIMILAR POLYGONS

  \dfrac{AH}{AC} = \dfrac{HK}{CE} = \dfrac{KB}{EF}

  If two lines are cut by any number of parallels, the corresponding intercepts are proportional.

• PROPORTION\@. SIMILAR POLYGONS

  \dfrac{AH}{m} = \dfrac{HK}{n} = \dfrac{KB}{p}

  The construction divides AB into parts AH, HK, KB proportional to the given lines m, n, p.

• PROPORTION\@. SIMILAR POLYGONS

  AG:AB = AB:AF

  In dividing a line in extreme and mean ratio, the whole line is the mean proportional between the segments on the chord construction.

Problems