Question

Three points A^ prime ,B and C^ prime are taken on the plane of a triangle ABC such that Delta A^ prime BC triangle AB^ prime C and triangle ABC^ prime are equilateral with their circumradii R_{a}, R_{b}, R_{c} inradii r_{a}, r_{b}, r_{c} and exradii r a ^ prime ,r b ^ prime ,r c ^ prime respec tively. Prove that (r a r b r c ):(R a R b R c ):(r a ^ prime r b ^ prime r^ prime )=1:8:27​

Answer 1

Answer:

Three points A^ prime ,B and C^ prime are taken on the plane of a triangle ABC such that Delta A^ prime BC triangle AB^ prime C and triangle ABC^ prime are equilateral with their circumradii R_{a}, R_{b}, R_{c} inradii r_{a}, r_{b}, r_{c} and exradii r a ^ prime ,r b ^ prime ,r c ^ prime respec tively. Prove that (r a r b r c ):(R a R b R c ):(r a ^ prime r b ^ prime r^ prime )=1:8:27

Answer 2

Answer:

Three points A^ prime ,B and C^ prime are taken on the plane of a triangle ABC such that Delta A^ prime BC triangle AB^ prime C and triangle ABC^ prime are equilateral with their circumradii R_{a}, R_{b}, R_{c} inradii r_{a}, r_{b}, r_{c} and exradii r a ^ prime ,r b ^ prime ,r c ^ prime respec tively. Prove that (r a r b r c ):(R a R b R c ):(r a ^ prime r b ^ prime r^ prime )=1:8:27