In this article, we will discuss the Circles and various theorem related to it. The circle is the most important topic of Geometry. In this article, I have covers tangents, chords and some important short trick also.

## CIRCLE:

- A circle is a set of point or locus of a point which are at a fixed distance from a point called as a centre.
- The distance of any point on the circumference of the circle from the centre of the circle is
**equal.**

**Diameter:**

Diameter is double of the radius .i.e. D=2R.

Diameter is the chord which passes through the centre of the circle.

**Tangent**: A line which touch the circle at only one point at its circumference.

**Secant**: A line which touch the circle at two distinct points.

**Chord**: A line Segment which lie inside the circle and its end points are always lie on the circle.

## Important theorem and results of the circle:

**(1) **Of the two chords of the circles, the one which is greater is** nearer** to the centre.

**(2) **The perpendicular from the centre of the circle **bisect** the chord i.e. radius always bisect the chord if perpendicular

**(3) A**ny line segment joining the centre of the circle and the mid point of the chord is perpendicular to the chord.

**If AM=MB then OM is perpendicular to the AB.(**reference to above fogure)

**(4)**Equal chord of the circle always subtends the equal angles at the centre of the circle.

**i.e. if AB=CD then ∠1=∠2**

**(5) **If angle subtended by the two chord at the centre are equal then the chords are always equal.

**(3) **Equal chords of the circle are at **equa**l distance from the centre.

**(6) C**hords which are at** equidistant** from the centre of the circle are always equal.

**(7)** Angle subtended by any arc at the centre of the circle is** double** the angle subtended by it at any point on the remaining part of the circle.

∠x=2∠y

**(8)** Angle subtended by an arc in the same segment of the circle are equal.

**∠ACB=∠ADB**

**(9)** Angle in a semi circle is a **right angle**. i.e the angle subtended by the diameter is always** right angle**.

**(10) **The circle drawn with **hypotenuse** of a right angle triangle as diameter, passes through its opposite vertex.

**(11)** The sum of the opposite angle of a cyclic quadrilateral is **always 180°**.

Cyclic quadrilateral is that quadrilateral whose all point lie on the circumference of the circle.

∠A+∠C=∠B+∠D

**(12)** If a side of a cyclic quadrilateral is produced then the exterior angle is **equal** to the interior opposite angle.

∠1=∠2

**Some important points of tangents**

**(1)**Tangent and radius always make the angle of 90 at the point of meeting of tangent with the circle.

**If AB is a tangent at P, then OP is perpendicular to AB.**

**(2) **The length of the two tangent drawn from the same external point to a circle is always equal.

**AP=AQ**

**(3) **If two chords AB and CD intersect internally or externally at point P then.

**PA*PB=PC*PD**

**(4)** If PAB is a secant which intersects the circle at A and B and PT be a tangent at T, then

**PT ^{2}=PA*PB**

**(5)** If from the point of contact of tangent with circle, a chord is drawn ,then the angles which the chord makes with the tangent line are equal respectively to the angles formed in the corresponding alternate segment.

**∠BAT=∠BCA=∠1**

**∠BAP=∠BDA=∠2**

**(6)** If two circles touch each other internally or externally the point of contacts lies on the line joining their centres.

Distance between the cntres

When touch internally, **distance=AP-BP**

When touch externally,** distance =AP+BP**

**Some Important Results:**

**(1)**If two tangent PA and PB are drawn from the external point P, then

∠1=∠2 and ∠3=∠4

OP is perpendicular to AB and AC=BC

**(2)** r1 and r2 are the radius of two circles and d is the distance between the centres of the circle then the length of the common tangent of two circles is given by

**(3)** If **r1** and **r2** are the two radius of the circle and **"d**" is the distance between them then the length of the transverse common tangent is given by

**(4) **If a circle touches all the four sides of a quadrilateral then the sum of opposite pair of sides are equal.

**i.e. AB+CD=AD+BC**

**(5) **If two chords AB and AC of a circle are equal then the bisector of ∠BAC passes through the centre O of the cirlcle.

**(6) **The equilateral formed by the angle bisector of a cyclic quadrilateral is also **cyclic**.

**(7)** If a cyclic** trapezium** is isosceles then its diagonals are equal

i.e if AB parallel DC and AD=BC then** AC=BD**

**(8)** Angle in the major segment of a circle is **acute** and angle in minor segment is **obtuse**.

**(9) **If two circles of same radius r are such that the centre of one lies on the circumference of the other then the length of the common chord is given by** l=√3*r**

**(10) **If **2a** and **2b** are length of two chords which intersects at right angle and if the distance between the centre of the circle and intersecting point of the chords is **C** then the radius of circle is given by]

**(11)** If three circles of radius** r** are bound by a rubber band then the length of rubber band is given by

**6r+2πr**

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