# 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.

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
*

*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*. A part of a circumference ( for instance, A

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

*tangent**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*,

*Fig.45 ),*

*based on the same arc*A

*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 ),
*
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 ! ).