Question

Prove Thales Theorem with statement

^_^

Answer 1

Your answer with statements is in the attachment:-


Hope it is helpful.........

Answer 2

Basic proportionality theorem (or) Thales theorem: If a line is drawn parallel to one side of a triangle intersecting the other two sides then it divides these sides in the same ratio.

Given: A Δ ABC. A line DE is parallel to BC, meeting AB at D and AC at E.

To prove: AD/DB = AE/EC

Proof: 

From the figure ,  ΔABC and  ΔADE we have 

 ∠ ABC =  ∠ ADE     (  Corresponding angles are equal.)

 ∠ ACB =  ∠ AED     (  Corresponding angles are equal.)

Then by AA axiom of similarity we can conclude that  Δ ABC   ΔADE 

We know that corresponding sides of a similar triangles are proportional. 
∴ AB/AD = AC/AE
          
 ⇒    AD + DB/AD
      
⇒  AE + EC/ AE     . . . . (  AB = AD + DB and AC = AE + EC)         

⇒ 1 + DB/AD
  =  1 +EC/AE

  ⇒ DB/AD = EC/AE

⇒ AD/DB = AE/EC     . . . (By taking reciprocals)   
 
      Hence,  AD/DB = AE/EC