In ` A B C` Prove that `A B^2+A C^2=2(A O^2+B O^2)` , where `O` is the middle point of `B C`

In A B C ,\ \ /_A=60o . Prove that B C^2=A B^2+A C^2-A B.A C .

Prove that b^2c^2+c^2a^2+a^2b^2gtabc(a+b+c) , where a,b,c gt 0 .

In o+ABC, prove that cos A+cos B+cos C<=(3)/(2)

In fig., O is a point in the interior of a triangle ABC, OD ⊥ BC, OE ⊥ AC and OF ⊥ AB. Show that:- O A^ 2 + O B^ 2 + O C^ 2 − O D^ 2 − O E^ 2 − O F^ 2 = A F^ 2 + B D^ 2 + C E^ 2

If A=2B, then prove that either c=b or a^(2)=b(c+b)

D is the mid-point of side B C of A B C and E is the mid-point of B Ddot If O is the mid-point of A E , prove that a r( B O E)=1/8a r( A B C)

If O is a point in space, A B C is a triangle and D , E , F are the mid-points of the sides B C ,C A and A B respectively of the triangle, prove that vec O A + vec O B+ vec O C= vec O D+ vec O E+ vec O Fdot

In figure, O is a point in the interior of a triangle ABC, O D_|_B C ,O E_|_A C and O F_|_A B . Show that (i) O A^2+O B^2+O C^2-O D^2-O E^2-O F^2=A F^2+B D^2+C E^2 (ii) A F^2+B D^2+C E^2=A W^2+C D^2+B F^2

In a quadrilateral A B C D , given that /_A+/_D=90o . Prove that A C^2+B D^2=A D^2+B C^2 .

Centroid of the tetrahedron OABC, where A=(a,2,3) , B=(1,b,2) , C=(2,1,c) and O is the origin is (1,2,3) the value of a^(2)+b^(2)+c^(2) is equal to: