# Geometrical locus. Circle and circumference

*Geometrical locus. Midperpendicular.*

Angle

bisector.

Circumference.

Circle. Arc. Secant. Chord. Diameter. Tangent line.

Segment of a circle. Sector of a circle. Angles in a circle.

Central angle. Inscribed angle. Circumscribed angle.

Radian measure of angles. Round angle. Ratio of

circumference length and diameter. Length of an arc.

Huygens’ formula. Relations between elements of a circle.

Angle

bisector.

Circumference.

Circle. Arc. Secant. Chord. Diameter. Tangent line.

Segment of a circle. Sector of a circle. Angles in a circle.

Central angle. Inscribed angle. Circumscribed angle.

Radian measure of angles. Round angle. Ratio of

circumference length and diameter. Length of an arc.

Huygens’ formula. Relations between elements of a circle.

** Geometrical locus **( or simply

**) is a totality of**

*locus**all*points, satisfying the certain given conditions.

E x a m p l e 1. A midperpendicular of any segment is a locus, i.e. a totality of all points,

equally

removed from the bounds of the segment. Suppose that PO

AB and

AO = OB :

Then, distances from any point P, lying on the midperpendicular PO, to bounds A and B of the segment AB are both equal to *d .*

So, *each point of a midperpendicular* has the following property: it *is removed from the bounds of the segment at equal distances.*

E x a m p l e 2. *
An
angle
bisector
is a
locus*,

that is a totality

of all points,

equally removed

from the angle sides.

that is a totality

of all points,

equally removed

from the angle sides.

E x a m p l e 3. A circumference is a locus, that is a totality of all points ( one of them – A ),

equally removed from its center O.

*Circumference** is a geometrical locus in a plane, that is a totality of all points, equally removed from its center. *Each of the equal

segments, joining the center with any point of a circumference is called a *radius* and signed as *r *or *R . A part of a plane
inside of a circumference, is called a circle*. A part of a circumference ( for instance, A

*m*B, Fig.39 ) is called an

*arc of a circle.*

The straight line PQ, going through two points M and N of a circumference, is called a

*secant*( or

*transversal*). Its segment MN, lying inside of the circumference,

is called a

*chord.*

A chord, going through a center of a circle ( for instance, BC, Fig.39 ), is called a *diameter* and signed as *d* or *D . *A diameter

is the greatest chord of a circle and equal to two radii ( *d *=* *2* r* ).

** Tangent. **Assume, that the secant PQ ( Fig.40 ) is going through points K and M of a circumference. Assume also, that point M is moving

along the circumference, approaching the point K. Then the secant PQ will change its position, rotating around the point K. As approaching the point M

to the point K, the secant PQ tends to some limit position AB. The straight line AB is called a

**or simply a**

*tangent line*

*tangent*to the circumference in the point K. The point K is called a

*point of tangency.*A tangent line and a circumference have only one common point – a

*point of tangency.*

*Properties of tangent. *

1) *A tangent to a circumference is perpendicular to a radius, drawing to a point of
tangency *( AB OK, Fig.40 )

*.*

2) *From a point, lying outside a circle, it can be drawn two tangents to the same
circumference; their segments lengths are equal* ( Fig.41 ).

** Segment of a circle **is a part of a circle, bounded by the arc ACB and the corresponding chord AB ( Fig.42 ). A length of the

perpendicular CD, drawn from a midpoint of the chord AB until intersecting with the arc ACB, is called a

*height*of a circle segment.

**is a part of a circle, bounded by the arc A**

*Sector of a circle**m*B and two radii OA and OB, drawn to the ends of the arc ( Fig.43 ).

** Angles in a circle. **A

*central angle*an angle, formed by two radii of the circle ( AOB,

**–**Fig.43 ). An

*inscribed angle*– an angle, formed by two chords AB and AC, drawn from one common point ( BAC,

Fig.44 ).

A *circumscribed angle* – an angle, formed by two tangents AB and AC, drawn from one common point ( BAC,

Fig.41 ).

** A length of arc **of a circle is proportional to its radius

*r*and the corresponding central angle :

*l =*

*r*

So, if we know an arc length *l *and a radius *r*, then the value of the corresponding central angle can

be determined as their ratio:

*= l / r .*

This formula is a base for definition of a *radian measure* of angles. So, if *l* = *r, * then

= 1, and we say, that an angle is equal to 1 radian (

it is designed as = 1 *rad* ). Thus, we have the following definition of a radian

measure unit: * A radian is a central angle* ( AOB, Fig.43 ), *whose arc’s length is equal to *

*its radius* ( AmB = AO, Fig.43 ). So, *a radian measure of any angle is a ratio of a length of an arc, drawn by an arbitrary radius and concluded
between the sides of this angle, to the radius of the arc. *Particularly, according to the formula for a length of an arc, a length of a circumference

*C*can be expressed as:

*r*,

where is determined as ratio of *C* and a diameter of a circle 2*r*:

*C /*2

*r .*

is an irrational number; its approximate value is 3.1415926…

On the other hand, 2 is a *round angle* of a circumference, which in a degree

measure is equal to 360 deg. In practice it often occurs, that both radius and angle of a circle are unknown. In this case, an arc length can be

calculated by the approximate Huygens’ formula:

*p*2

*l*+ ( 2

*l – L*) / 3 ,

where ( according to Fig.42 ): *p* – a length of the arc ACB; *l* – a length of the chord AC; *L* – a

length of the chord AB. If an arc contains not more than 60 deg, a relative error of this formula is less than 0.5%.

*Relations between elements of a circle. ** An inscribed angle * ( *
ABC*,

*Fig.45*

*)*

*is equal to a half of the central angle*(

*,*

AOC

AOC

*Fig.45 ),*

*based on the same*

arcA

arc

*m*C. Therefore,

*all inscribed angles*( Fig.45 ),

*based on the same arc*( A

*m*C, Fig.45 ),

*are equal.*

As a central angle contains the same quantity of degrees, as its arc ( A

*m*C, Fig.45 ), then any

*inscribed angle is measured by*

*a half of an arc, which is based on*( A

*m*C in our case ).

*All inscribed angles, based on a semi-circle* (APB, AQB, …,

Fig.46 ), * are right angles* ( Prove this, please ! ). * An angle *(AOD,

Fig.47 ),* formed by two chords *( AB and CD ), *is measured by a semi-sum of arcs, concluded between its sides:*( A

*n*D + C

*m*B ) / 2 .

*An angle *(AOD, Fig.48 )*, formed by two secants *( AO and OD ), *is measured
by a semi-difference of arcs, concluded between its sides: *( A

*n*D – B

*m*C

*) / 2 .*

*An angle*(DCB, Fig.49 )

*, formed by a tangent and a chord*( AB and CD ),

*is measured by a half of an arc, concluded inside of it:*C

*m*D / 2 .

*An angle*(BOC, Fig.50 )

*,*

formed by a tangent and a secant( CO and BO ),

formed by a tangent and a secant

*is measured by a semi-difference of arcs, concluded between its sides:*

( B

*m*C

*–*C

*n*D

*) / 2 .*

*A circumscribed angle *(AOC, Fig.50 )*, formed by the two tangents, *(CO and AO),

*is measured by a semi-difference of arcs, concluded between its sides: * ( ABC* – *CDA ) / 2 *.*

*Products of segments of chords *( AB and CD, Fig.51 or Fig.52 ), *into which
they are divided by an intersection point, are equal: *AO · BO = CO · DO.

A*
square
of tangent line
segment
is equal
to
a
product
of a secant line segment by the secant line
external part
* ( Fig.50 ):

OA

^{2 }

= OB

· OD (

prove, please!

).

This property may be considered as a particular case of Fig.52.

*A chord *( AB, Fig.53 )*, which is perpendicular to a diameter *( CD )*, is divided into two in the intersection point *O :

AO = OB *.
*( Try to prove this ! ).