Theorems on Chord
In this section we will discuss Theorems on Chord.Theorems on Circles at both the Intermediate and Higher tier are difficult areas for students, which teachers often find equally difficult to deliver.Here, I have explained some Circle theorems based on circle so that student can understand them easily.
1) If two arcs of a circle are congruent, then corresponding chords are equal.
Given : PQ = BC
Prove that : ∠POQ = ∠BAC


1) BC = PQ  1) Given 
2) OP = AB  2) Radius of a circle 
3) OQ = AC  3) Radius of a circle 
4) ΔPOQ = ΔBAC  4) SSS Postulate 
5) ∠POQ = ∠BAC  5) CPCTC 
Given : PQ is a chord of a circle and OL ⊥ PQ.
Prove that : LP = LQ


1) OP = OQ  1) Radii of the same circle. 
2) OM = OM  2) Reflexive (common) 
3) ∠OMP = ∠OMQ  3) Each 90^{0} 
4) ΔPMO = ΔQMO  4) HL postulate(RHS) 
5) PM = MQ  5) CPCTC 
Given : AB = AC
Prove that : Center O lies on the bisector of ∠BAC


1) AB = AC  1) Given 
2) ∠BAM = ∠CAM  2) Given 
3) AM = AM  3) Reflexive (Common) 
4) ΔBAM = ΔCAM  4) SAS Postulate 
5) BM = CM and ∠BMA = ∠CMA  5) CPCTC 
6) ∠BMA + ∠CMA = 90^{0}  6) Linear pair angles 
7) AM = BM and ∠BMA = ∠CMA = 90^{0}  7) From (6) 
8) AM is the perpendicular bisector of BC  8) Definition of perpendicular bisector. 
9) AM passes through the center O.  9) Perpendicular bisector of a chord always passes through the center. 
Theorems on Circle
• Theorems on Chord
• Theorems on Chord and Subtended Angle
• Theorems on Arc and Angle
• Theorems on Cyclic Quadrilateral