THE CIRCLE

Excerpts

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  In the same circle or in equal circles, two central angles have the same ratio as their intercepted arcs.

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  An inscribed angle is measured by half the arc intercepted between its sides.

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  A circumference is divided into $360$ equal parts, called *degrees*; and therefore a unit angle at the centre intercepts a unit arc on the circumference.

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  An angle formed by two chords intersecting within the circumference is measured by half the sum of the intercepted arcs.

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  An angle inscribed in a semicircle is a right angle.

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  A **circle** is a portion of a plane bounded by a curved line, all points of which are equally distant from a point within called the **centre**.

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  Two quantities of the same kind that cannot *both* be expressed in *integers* in terms of a common unit, are said to be **incommensurable**, and the *exact value* of their ratio cannot be found.

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  If, however, a theorem is in fact a group of three theorems, and if *one of the hypotheses* of the group *must* be true, and *no two of the conclusions can be true at the same time*, then the converse of the theorem is *necessarily* true.

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  To **measure** a quantity of any kind is to find *the number of times* it contains a known quantity of the *same kind*, called the **unit of measure**.

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  A straight line perpendicular to a radius at its extremity is a tangent to the circle.

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  By the definition of a circle, *all its radii are equal*. All its diameters are equal, since a diameter is equal to two radii.

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  A **tangent** is a straight line of unlimited length which has one point, and only one, in common with the circumference; as, $BC$ (Fig. 1).

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  No quantity is great or small except by comparison with another quantity of the *same kind*.

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  But by taking the unit sufficiently small, an *approximate value* can be found that shall differ from the true value of the ratio by less than any assigned value, however small.

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  If a variable, by having different successive values, can be made to differ from a given constant by less than any assigned value, however small, but cannot be made absolutely equal to the constant, that constant is called the **limit** of the variable, and the variable is said to **approach the constant as its limit**.

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  Then it is evident that the moving point *may approach as near to $B$ as we choose, but will never arrive at $B$*.

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  We cannot make $DB'$ equal to zero, since, by hypothesis, $AB$ and $A'B'$ are incommensurable.

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  Thus, if $OA$ is considered positive, then $OC$ may be considered negative, and if $OR$ is considered positive, then $OD$ may be considered negative.

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  By marking the distinction between quantities measured in opposite directions, a theorem may often be so stated as to include two or more particular theorems.

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  An angle included by a tangent and a chord drawn from the point of contact is measured by half the intercepted arc.

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  Here the word *sum* means the algebraic sum and includes both the arithmetical sum and the arithmetical difference of two quantities.

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  Prove that the locus of the vertex of a right triangle, having a given hypotenuse as base, is the circumference described upon the given hypotenuse as diameter (§ 290).

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  The required point is the intersection of the given line with the perpendicular bisector of the line joining the two given points (§ 160).

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  Make use of the point which forms with $P$ a pair of points symmetrical with respect to $AB$.

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  Let $r$ and $r'$ denote the radii of the circles, $O$ and $O'$ their centres.

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  Let $ABC$ be the $\triangle$ required, $EF$ the given perimeter. The altitude $CD$ passes through the middle of $EF$, and the $\triangle_s AEC$, $BFC$ are isosceles.

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  To bisect the angle formed by two lines, without producing the lines to their point of intersection.

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  To draw the internal tangents use an auxiliary $\odot$ of radius $r + r'$.

Equations

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  \dfrac{a}{b}

  The ratio of a to b is written a : b, or as the fraction a/b.

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  \frac{3}{10} + \frac{3}{100} + \frac{3}{1000} + \cdots

  The decimal 0.333... is written as the infinite sum of the fractions 3/10, 3/100, 3/1000, and so on, whose sum approaches 1/3 as a limit.

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  \sqrt{2} = 1.41421356\cdots

  The square root of 2 is an incommensurable value whose decimal expansion is given to eight places, so approximate values can be taken from it.

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  \dfrac{x}{k} = \dfrac{1}{k} × x

  Dividing a variable by a finite constant k is the same as multiplying it by 1/k.

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  kx=0

  If the limit of the variable x is zero, then the limit of kx, for any finite constant k, is zero.

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  kx = ka

  The limit of kx equals k times the limit a of x, for any finite constant k.

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  xy = ab

  The limit of the product xy of two variables is the product ab of their respective limits, provided neither limit is zero.

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  x^n = a^n

  The limit of the nth power of a variable x is the nth power a^n of its limit a.

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  d+d'+d''+\cdots < nd

  The sum of n differences is less than n times the largest difference d, which is the step used to show that the sum of the differences can be made smaller than any assigned quantity.

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  \dfrac{a}{b} = r

  If the variables x and y have a constant ratio r and their limits a and b are not zero, then the ratio of the limits a/b equals r.

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  AB = AC

  The two tangents drawn from an external point A to a circle are equal in length.

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  \dfrac{\angle A'C'B'}{\angle ACB} = \dfrac{\arc A'B'}{\arc AB}

  In equal or the same circle, the ratio of two central angles equals the ratio of their intercepted arcs.

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  \angle ECF = 90° + \frac{1}{2}\angle ACB

  The angle ECF in the analysed triangle equals a right angle plus half the given angle ACB.

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  \angle E+\angle F+\frac{1}{2}\angle ACB = 90°

  The angles at E and F of triangle ECF, plus half the given angle ACB, sum to a right angle.

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  \angle E+\angle F = 90° - \frac{1}{2}\angle ACB

  The sum of the angles at E and F equals a right angle minus half the given angle ACB.

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  EG = FC = \frac{1}{2}DC

  In the isosceles trapezoid construction, the segments EG and FC are each half of the base DC, as the proof states.

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  \angle BAC = \angle BCA = 45°

  In the square construction, the angles BAC and BCA each equal 45°, because the triangles ABC and ABE are isosceles and CA is a diagonal of the square.

Problems