Q. 144.3( 11 Votes )

If is isosceles with AB = AC and C (O, r) is the incircle of the touching BC at L, prove that L bisect BC.

Answer :

Given: If is isosceles with AB = AC and C (O, r) is the incircle of the touching BC at L.

To prove: L bisect BC.

Proof: 

Construct the figure according to given condition.




AB = AC (given)

From the theorem which states that the lengths of two tangents drawn from external point to a circle are equal.  ..... (1)

As tangents AP and AQ are drawn from the external point A.

AP = AQ  

Also,

AB = AC

⇒ AP + PB = AQ + QC

⇒ AP + PB = AP + QC

⇒ PB =  QC


From (1) as tangents BP and BL are drawn from external point B,

And tangents CQ and CL are drawn from external point C.

⇒ BP = BL  ..... (3)

   CQ = CL ...... (4)

As we have proved PB = QC

From 3 and 4

BL = CL

⇒ L bisects BC.

Hence proved.

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Quiz | Imp. Qs. on CircleQuiz | Imp. Qs. on CircleQuiz | Imp. Qs. on Circle35 mins
Quiz | Imp. Qs. on CirclesQuiz | Imp. Qs. on CirclesQuiz | Imp. Qs. on Circles37 mins
Quiz | Testing Your Knowledge on CirclesQuiz | Testing Your Knowledge on CirclesQuiz | Testing Your Knowledge on Circles32 mins
Short Cut Trick to Find Area of TriangleShort Cut Trick to Find Area of TriangleShort Cut Trick to Find Area of Triangle43 mins
Quiz | Areas Related to CirclesQuiz | Areas Related to CirclesQuiz | Areas Related to Circles43 mins
RD Sharma |  Area of Sector and SegmentsRD Sharma |  Area of Sector and SegmentsRD Sharma | Area of Sector and Segments25 mins
Quiz | Area Related with CirclesQuiz | Area Related with CirclesQuiz | Area Related with Circles47 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses