Use app×
Join Bloom Tuition
One on One Online Tuition
JEE MAIN 2025 Foundation Course
NEET 2025 Foundation Course
CLASS 12 FOUNDATION COURSE
CLASS 10 FOUNDATION COURSE
CLASS 9 FOUNDATION COURSE
CLASS 8 FOUNDATION COURSE
0 votes
6.7k views
in Mathematics by (30.0k points)

Bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and F respectively. 

Prove that the angles of the triangle DEF are 

90° – 1/ 2 A, 90° – 1/ 2 B and 90° – 1/ 2 C

1 Answer

0 votes
by (130k points)
selected by
 
Best answer

Given : ∆ABC and its circumcircle. AD, BE, CF are bisectors of ∠A, ∠B, ∠C respectively. 

Construction : Join DE, EF and FD. 

Proof : We know that angles in the same segment are equal. 

∴ ∠5 = ∠C 2 and ∠6 = ∠B 2 ..(i) 

∠1 = ∠A 2 and ∠2 = ∠C 2 ..(ii) 

∠4 = ∠A 2 and ∠3 = ∠B 2 ..(iii) 

From (i), we have 

∠5 + ∠6 = ∠C/ 2 + ∠B/ 2 

⇒ ∠D = ∠C/ 2 + ∠B/ 2 ...(iv) 

[∵∠5 + ∠6 = ∠D] 

But ∠A + ∠B + ∠C = 180° 

⇒ ∠B + ∠C = 180° – ∠A

Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students.

Categories

...