If `8R^2=a^2+b^2+c^2` then show that the triangle is right angled

If in a triangle ABC, cos^(2)A + cos^(2)B + cos^(2)C =1 , then show that the triangle is right angled.

In any triangle ABC, if 8R^(2) =a^(2) + b^(2) +c^(2) , prove that, the triangle is right angled.

Let ABC be a triangle having O and I as its circumcentre and incentre, respectively. If R and r are the circumradius and the inradius respectively, then prove that (IO) 2 =R 2 −2Rr. Further show that the triangle BIO is right angled triangle if and only if b is the arithmetic mean of a and c.

If in a triangle A B C , (2cosA)/a+(cos B)/b+(2cosC)/c=a/(b c)+b/(c a) , then prove that the triangle is right angled.

Let alt=blt=c be the lengths of the sides of a triangle. If a^2+b^2 < c^2, then prove that triangle is obtuse angled .

If (a^(2) + b^(2))sin (A-B) = (a^(2)-b^(2)) sin(A+B) then show that, the triangle is either isoscelels or right angled.

A(2, 3, 1), B(-2, 2, 0) and C(0, 1, -1) are the vertices of the triangle ABC. Show that the triangle ABC is right -angled.

If r_1 =r_2+r_3+r, prove that the triangle is right angled.

If sinA=sin^2Ba n d2cos^2A=3cos^2B then the triangle A B C is right angled (b) obtuse angled (c)isosceles (d) equilateral

(i) If the cosines of two of the angles of a triangle, which is not isosceles, are iversely proportional to the corresponding opposite sides, show that the triangle is right angled. (ii) In a triangle ABC, If A=2B then show that, a^(2)- b^(2)=bc .