DE is parallel to OB, EF Is parallel to BC prove that DF parallel OC

Dear student
In ABO, we haveDEOBBy Basic proportionality Theorem, we haveAEEB=ADDO     ...(1)In ABC, we haveEFBCBy Basic proportionality Theorem, we haveAEEB=AFFC     ...(2)From (1) and (2), we haveADDO=AFFCDFOC  By the converse of Basic proportionality Theorem 

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We have given
1. D E is parallel to O B
​2. E  F is parallel to BC

To prove:- D F is parallel to O C
Proof:- In the given question we produce E D to H and B O to G and produce A O to 
intersect B C at M. We also produce F D to N and CO to P.
Now, angle H D F = angle M D E (vertically opposite angles)
         angle G O C = angle P O B
Now, since angle M D E = angle P O B (Since E D is parallel to B O and A M is transversal)
Hence angle H D F = angle G O C
Now, angle A D H = angle DOG
Now, 180-(angle A D H+angle H D F)=180-(angle G O C+angle G O D )
 angle M D F=angle M O C                                                               1
 D F is parallel to O C (if we consider AM a transversal then from 1) 
Thus proved.
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