Prove that `[axxb, a xx c, d] = (a*d)[a, b, c]`

If a, b, c and d are four coplanr points, then prove that [a b c]=[b c d]+[a b d]+[c a d] .

If a ,b,c , d are in G.P. prove that (b + c)(b + d) = (c + a)(c+d)

Value of [vec a xx vec b,vec a xx vecc,vec d] is always equal to (a) ( ⃗ a . ⃗ d ) [ ⃗ a ⃗ b ⃗ c ] (b) ( ⃗ a . ⃗ c ) [ ⃗ a ⃗ b ⃗ d ] (c) ( ⃗ a . ⃗ b ) [ ⃗ a ⃗ b ⃗ d ] (d) none of these

In A B C , ray A D bisects /_A and intersects B C in D . If B C=a , A C=b and A B=c , prove that B D=(a c)/(b+c) (ii) D C = (a b)/(b+c)

If A B C D is a parallelogram, the prove that a r( A B D)=a r( B C D)=a r( A B C)=a r( A C D) =1/2a r (||^(gm) A B C D)

In Figure, A C=B C ,\ /_D C A=/_E C B\ a n d\ /_D B C=E A Cdot Prove that triangle D B C\ a n d\ E A C are congruent, and hence D C=E C\ a n d\ B D=A E

If a + c = b e and 1/b + 1/d = e/c , Prove that : a,b,c and d are in proportion.

If a ,b ,c ,d are in G.P., prove that a+b,b+c ,c+d are also in G.P.

If vec a , vec b , vec ca n d vec d are the position vectors of the vertices of a cyclic quadrilateral A B C D , prove that (| vec axx vec b+ vec bxx vec d+ vec d xx vec a|)/(( vec b- vec a)dot( vec d- vec a))+(| vec bxx vec c+ vec cxx vec d+ vec d xx vec b|)/(( vec b- vec c)dot( vec d- vec c))=0dot

Diagonals A C\ a n d\ B D of a trapezium A B C D with A B||D C intersect each other at O . Prove that a r\ ( A O D)=\ a r\ ( B O C) .