# If two points on a given line subtend equal angles at two distinct points which lie on the same side of the line, then the four points are

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ago in Circle
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Theorem :  If two points on a given line subtend equal angles at two distinct points which lie on the same side of the line, then the four points are concyclic. +1 vote
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Given: Points B and C lie on the same side of the line AD ∠ABD = ∠ACD

To prove: Points A, B, C, D are concyclic.

i.e., □ABCD is a cyclic quadrilateral.

Proof:

Suppose points A, B, C, D are not concyclic points.

We can still draw a circle passing through three non collinear points A, B, D.

Case I: Point C lies outside the circle. Then, take point E on the circle such that A – E – C.

∠ABD ≅ ∠AED (i) [Angles inscribed in the same arc]

∠ABD ≅ ∠ACD (ii) [Given]

∴ ∠AED ≅ ∠ACD [From (i) and (ii)]

∴ ∠AED ≅ ∠ECD [A – E – C] But, ∠AED ≅ ∠ECD as ∠AED is the exterior angle of ∆ECD.

∴ Our supposition is wrong.

∴ Points A, B, C, D are concyclic points.

Case II: Point C lies inside the circle. Then, take point E on the circle such that A – C – E.

∠ABD ≅ ∠AED (iii) [Angles inscribed in the same arc]

∠ABD ≅ ∠ACD (iv) [Given]

∴ ∠AED ≅ ∠ACD [From (iii) and (iv)]

∴ ∠CED ≅ ∠ACD [A – C – E] But, ∠CED ≅ ∠ACD as ∠ACD is the exterior angle of ∆ECD.

∴ Our supposition is wrong.

∴ Points A, B, C, D are concyclic points.