In the given figure, two circles intersect at M and N. ABC is a triangle. State whether the statement is true or false:

A,L,M, and K are concyclic points.

Question

In the given figure, two circles intersect at M and N. ABC is a triangle. State whether the statement is true or false:A,L,M, and K  are concyclic points.

Accepted Answer

It is given that; two circles intersect at M and N and ABC is a triangle.We need to find if A,L,M, and K  are concyclic points is true or false.We know that a cyclic quadrilateral's opposing angles are supplementary. That is, the total of the opposite angles equals

180
    °

.The exterior angle of a cyclic quadrilateral equals the interior opposite angle.In the given figure, join MN,LM,KM . Observe that LMNB lie on the same circle.So, LMNB is a cyclic quadrilateral. KMNC is also a cyclic quadrilateral.

∠1+∠2=
    
     180
    °

(Property of cyclic quad.) ...(i)and

∠3+∠4=
    
     180
    °

(Property of cyclic quad.) ...(ii)

∠1=∠4
   (exterior angle property of cyclic quadrilateral) ...(iii) and

∠3=∠2
   (exterior angle property of cyclic quadrilateral) ...(iv)We can observe that, BNC is a line segment and so, ﻿

∠2+∠4=
    
     180
    °

.Solve equations (iii) and (iv).

⇒∠1=∠3=∠2=∠4=
    
     90
    °
   
   …(v)
  By the rule of linear pair angles,

⇒∠1+∠MLA=
        
         180
        °

⇒∠3+∠MKA=
        
         180
        °

Now, use equation (v) and solve further.

⇒∠MLA=∠MKA=
    
     90
    °

The sum of supplementary angles is

180
    °

.

∠MLA+∠MKA=
    
     180
    °

Therefore, LMKA is a cyclic quadrilateral. So, A,L,M, and

K
   are concyclic points.Given statement is true.The correct option is 1).

In the given figure, two circles intersect at M and N. ABC is a triangle. State whether the statement is true or false:

A,L,M, and K are concyclic points.

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