If the lengths of the sides `vec(BC), vec(CA)` and `vec(AB)` of the triangle ABC be a:b:c respectively and the lengths of perpendicular from the circumcenter upon `vec(BC), vec(CA)` and `vec(AB)` be x, y, z respectively, then prove that,
`a/x + b/y + c/z = (abc)/(4xyz)`.

ABC is a right angled triangle , right angled at C and P is the length of perpendicular from C on AB. If a, b and C are the length of sides BC, CA and AB respectively. Then___

If the vectors vec a , vec b ,a n d vec c form the sides B C ,C Aa n dA B , respectively, of triangle A B C ,t h e n

The perpendicular from the vertices A, B and C of a triangle ABC on the opposite sides meet at O. If OA=x, OB=y and OC=z, prove that, a/x + b/y + c/z = (abc)/(xyz) .

If vec (a), vec(b) and vec ( c) are three vectors of magnitude 3,4 and 5 respectively such that each vector is perpendicular to the sum of the other two vectors , prove that | vec (a) + vec(b) + vec (c ) | = 5 sqrt (2)

Prove (a + b + c) (ab + bc + ca) > 9abc

Let D, E, F are the midpoints of the sides bar(BC), bar(CA) " and " bar(AB) of the triangle ABC. Prove that vec(AD)+vec(BE)+vec(CF)=vec(0) .

If vec a+ vec b+ vec c = vec 0 then prove that vec axx vec b= vec bxx vec c = vec cxxvec a .

D, E, F are the midpoints of the sides bar(BC), bar(CA) " and " bar(AB) respectively of the triangle ABC. If P is any point in the plane of the triangle, show that vec(PA)+vec(PB)+vec(PC)=vec(PD)+vec(PE)+vec(PF) .

Suppose that vec(p), vec(q) and vec(r) are three non-coplanar vectors in R. Let the components of a vector vec(s) along vec(p), vec(q) and vec(r) be 4, 3 and 5 respectively. If the components of this vector vec(s) " along" (-vec(p) + vec(q) + vec(r)), (vec(p) - vec(q) + vec(r)) and (-vec(p) - vec(q) + vec(r)) are x,y and z respectively, then the value of 2x + y + z is

If the length of the sides of the triangle ABC satisfy, 2(bc^2+ca^2+ab^2)=b^2c+c^2a+a^2b+3abc , then triangle ABC is: