Prove without expansion that `|a h+bgga b+c h bf+b afh b+b c af+b cc bg+fc|=a|a h+bga h bf+b a h b af+b cgf|`

Prove without expansion that |(ah+bg, g, ab+ch) (bf+ba, f, hb+bc) (af+bc, c , bg+fc)|=a|(ah+bg ,a, h) (bf+ba, h, b) (af+bc, g, f)|

Prove without expansion that |{:(a h+bg,g,ab+ch), (bf+ba,f,hb+bc), (af+bc,c,bg+fc):}|=a|{:(ah+bg,a, h), (bf+ba,h,b),(af+bc,g,f):}|

If a : b = c : d, and e : f = g : h, then (ae + bf ) : (ae – bf ) =?

If b +c, c + a, a + b are in H.P then

If a , b , c are in H.P. , then

f is a continous function in [a, b] ; g is a continuous function in [b,c]. A function h(x) is defined as h(x)=f(x) for x in [a,b) , g(x) for x in (b,c] if f(b) =g(b) then (A) h(x) has a removable discontinuity at x = b. (B) h(x) may or may not be continuous in [a, c] (C) h(b-) = g(b+ ) and h(b+ ) = f(b- ) (D) h(b+ ) = g(b- ) and h(b- ) = f(b+ )

Prove that if a,b,c are in H.P. if and only if (a)/c = (a-b)/(b-c)

If a ,b ,c in R^+, t h e n(b c)/(b+c)+(a c)/(a+c)+(a b)/(a+b) is always