In any triangle, show that the perpendicular bisectors of the sides are concurrent.
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
Let ABC be the triangle and D, E and F are respectively middle points of sides BC, CA and AB. Let the perpendicular bisectors of BC and CA meet at O. Join OF. We are required to 
prove that OF is ⊥ to AB. Let the position vectors of A, B, C with O as origin of reference be 
, 
and 
respectively.
∴ 
= 
( 
+ 
), 
= 
( 
+ 
) and 
= 
( 
+ 
)
Also 
= 
– 
, 
= 
– 
and 
= 
– 
Since OD ⊥ BC
⇒ 
( 
+ 
) . ( 
– 
) = 0 ⇒ b 2 = c 2 ............(i)
Similarly OE ⊥ CA ⇒ 
( 
+ 
) . ( 
– 
) = 0 ⇒ a 2 = c 2 ...........(ii)
from (i) and (ii) we have a 2 – b 2 = 0
⇒ ( 
+ 
) . ( 
– 
) = 0 ⇒ 
( 
+ 
) . ( 
– 
) = 0 ⇒ 
.
──────────────────────────────────────────────────────────────────────────────────────────
Prepare Smarter with CGP Edu
Get practice questions, solutions, and test series in one place.
Write a Review
Share your experience with this question and solution.
Commentary
Send your comment, doubt, correction, or feedback to admin.
Similar Questions
Explore conceptually related problems