If `8R^(2)=a^(2) +b^(2) +c^(2).` then prove that the `Delta` is right angled.

In any triangle. if(a^2-b^2)/(a^2+b^2)=("sin"(A-B))/("sin"(A+B)) , then prove that the triangle is either right angled or isosceles.

ABC is an isosceles triangle, with AC= BC. If AB^(2)= 2AC^(2) , prove that ABC is a right triangle.

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

Show the complex numbers -2+3i,-2i and 4-I in the Argand plane. Prove that they are vertices ofa right angle triangle.

If A,B and C are the angles of a triangle and |{:(1,1,1),(1+sinA,1+sinB,1+sinC),(sinA+sin^(2)A,sinB+sin^(2)B,sinC+sin^(2)C):}| =0 then prove that Delta ABC must be isoceles.

Without using Pythagoras theorem prove that A(12, 8), B(-2, 6) and C(6, 0) are the verticies of right angled triangle.

The vertices of a triangle are (1, a),(2, b) and (c^2,-3). (1) Prove that its centroid can not lie on the y-axis. (2) Find the condition that the centroid may lie on the x-axis for any value of a,b,c in R

Complex numbers z_(1),z_(2)andz_(3) are the vertices A,B,C respectivelt of an isosceles right angled triangle with right angle at C. show that (z_(1)-z_(2))^(2)=2(z_1-z_(3))(z_(3)-z_(2)).

If the points A(-2, k), B(3,-4) and C(7,10) are the vertices of a right angled isosceles triangle right angled at A, find the value of k and the area of Delta ABC .

If A(2,2) , B(-4,-4) and C(5,-8) are the vertices of Delta ABC then the length of the median through C is . . . . . Units