In a triangle ABC, vertex angle A, B, C and side BC are given. The area of triangle ABC is If cot A/2 = (b+c)/a, then triangle ABC is In triangle ABC, tan A/2 = 5/6, tan C/2 = 2/5,... In a triangle ABC, vertex angle A, B, C and side BC are given. The area of triangle ABC is If cot A/2 = (b+c)/a, then triangle ABC is In triangle ABC, tan A/2 = 5/6, tan C/2 = 2/5, then a, b, c are in A.P. In a triangle ABC, the line joining the circumcentre to the incentre is parallel to BC, then cos B + cos C = In triangle ABC, r = Read more

Steps to Solve

1. Understanding the given statements The problem consists of several statements regarding the properties and relations in triangle ABC. Each statement can lead to a conclusion about the triangle's characteristics, specifically focusing on angles and side lengths in relation to each other.

2. Analysis of cotangent in the triangle The statement "If $\cot \frac{A}{2} = \frac{b+c}{a}$" suggests a specific relationship. Using the cotangent half-angle formula, this gives insight into the triangle's balance of sides and angles.

3. Investigation of the triangle type From the equation $\cot \frac{A}{2} = \frac{b+c}{a}$:

   • If $A$ is an angle, and sides $b$ and $c$ are equal (when the triangle is isosceles), then this equation holds true, indicating that triangle ABC is isosceles.

4. Verifying potential types of triangle To confirm, we need to check:

   • Equilateral: All sides equal, condition not satisfied.
   • Right angled: The angle A could possibly be a right angle, but it doesn't satisfy the equality since the relation is specifically tied to the sides.
   • Conclusion: Triangle ABC must then be isosceles.