Question

ABC is a triangle with AB=AC,D and E are points on the sides AB and AC respectively,such that AD=AE.show thatB,C,D andE are concyclic​

Answer 1

Step-by-step explanation:

Given:

AD = AE …(i)  

AB = AC …(ii)  

Subtracting AD from both sides, we get:  

⟹ AB – AD = AC – AD  

⟹ AB – AD = AC - AE (Since, AD = AE)  

⟹ BD = EC …(iii)  

Dividing equation (i) by equation (iii), we get:

AD/DB = AE/EC

Applying the converse of Thales’ theorem, DE‖BC  

⟹ ∠DEC + ∠ECB = 1800 (Sum of interior angles on the same side of a Transvesal Line is 00 .)  

⟹ ∠DEC + ∠CBD = 1800 (Since, AB = AC ⟹ ∠B = ∠C)  

Hence, quadrilateral BCED is cyclic.  

Therefore, B,C,E and D are concylic points.