In this article we have made a collection of 10 important circle theorems that you must know if you are a student or someone who wants to master everything about circle. Having read these theorems you will feel quite confident in solving the circle problems that you are often stuck on.
To understand these circle theorems you just need to have a basic idea of Circle fundamentals and triangle congruence and similarity. We first learn about the theorem followed by the proof for it. If you are not good at fundamentals of circle you can refer to another on this portal that is basically dedicated to the Basics of Circle . So let's start with them one by one.
10 Circle theorems to get a hold on Geometry
To comprehend the nuances of mathematics we should focus on spending most of our time on concept building rather than starting with problem-solving.
Theorem 1 : Any line passing through the centre of the circle is always perpendicular to the tangent at the point of contact.
Theorem 2 : Tangents drawn to the circle from a point outside the circle are always of equal in length.
Proof : we have to prove that AM = AN
Let's consider a point A outside the circle and draw two imaginary lines passing through the point A that touches the circle at two points M and N. O be the centre of the circle
Line AM and AN are tangents to the circle and we will prove they are equal.
Now consider the \(\Delta OMA \) and \(\Delta ONA \)
From theorem 1. AM is tangent to the circle hence \( \angle OMA \) and \( \angle ONA \) are \( 90°\)
\( \angle OMA\) =\( \angle ONA \) (Both 90°)
OA = OA ( common side and Hypotenuse in both the triangles )
OM = ON ( Radius of the circle)
By RHS congruence rule both the triangles
\(\Delta OMA \) and \(\Delta ONA \) are proven to be congregent.
\( \implies \) AM = AN
Theorem 3 : If a line passes through the centre and perpendicular to any chord on the circle it bisects the chord.
Proof : We have to prove AD = DB
Drop an imaginary perpendicular line from centre O to the chord AB. The perpendicular touches the chord at point D.
Now look at \(\Delta OAD \) and \(\Delta ODB \)
OA = OB ( Both are radius)
OD =OD ( common side in both triangles)
\( \angle ODA = \angle ODB \)
By RHS congruence rule \(\Delta OAD \) and \(\Delta ODB \) are congruent.
\(\therefore \) AD = DB (Both sides are equal)
Theorem 4 : Angle subtended by an arc at the centre is always double of the measure of the angle subtended by the same arc at any other point of the remaining part of the circle.
Proof : To Prove \( \angle ACB = 2\angle AOB \)
Theorem 5 : Angles formed on the same segment are always equal in measure.
Proof : Let AB is the chord and \(\Delta ACB \) and \(\Delta ADB \) are the triangles formed on the same chord AB.
we will prove \( \angle ACB \) = \( \angle ADB \)
From \(\Delta ACB \)
\( \angle ACB + \angle CAB + \angle CBA \) = 180° ..............(Eq_1)
\( \angle ADB + \angle DAB + \angle DBA \) = 180° ..............(Eq_2)
Eq_(1) = Eq_(2)
From the diagram we can observe that
\(\angle CAB = \angle CAD + \angle DAB \)
\(\angle CBA = \angle CBD + \angle DBA \)
Theorem 6 : If two chords of a circle are equal in length they subtend equal angles at the centre of the circle.
Theorem 7 : If two chords subtend equal angles at the centre they are of equal length.
Theorem 8 : If two chords are equidistant from the centre, they are always equal in length.
Theorem 9 : If two chords are equal in length they are always of equal length.
Theorem 10 : A diameter of a circle subtends a right angle at the circumference of the circle.