`(a^(2)-b^(2)-c^(2)+d^(2))^(2)-4(ad-bc)^(2)`

If (a)/(b)=(c)/(d) then (a+b)/(a-b)=(c+d)/(c-d),ab=cd ,(a^(2)+b^(2))/(a^(2)-b^(2))=(c^(2)+d^(2))/(c^(2)-d^(2)), ad-dc=0

If a,b,c,d,e are in continued proportion,then prove that (ab+bc+cd+de)^(2)=(a^(2)+b^(2)+c^(2)+d^(2))(b^(2)+c^(2)+d^(2)+e^(2))

If D is the mid-point of the side BC of triangle ABC and AD is perpendicular to AC, then 3b^(2)=a^(2)-c( b )3a^(2)=b^(2)3c^(2)b^(2)=a^(2)-c^(2)(d)a^(2)+b^(2)=5c^(2)

If a,b,c,d are in G.P.prove that: (a^(2)+b^(2)),(b^(2)+c^(2)),(c^(2)+d^(2)) are in G.P.(a^(2)-b^(2)),(b^(2)-c^(2)),(c^(2)-d^(2)) are in G.P.(1)/(a^(2)+b^(2)),(1)/(b^(2)+c^(2)),(1)/(c^(2)+d^(2)) are in G.P.(a^(2)+b^(2)+c^(2)),(ab+bc+cd),(b^(2)+c^(2)+d^(2))

If a:b=b:c,(:thenbackslash a^(4):b^(4) would be equal to ac:b^(2) b.a^(2):c^(2) c.c^(2):a^(2) d.b^(2):ac

Prove that Delta=|(1,bc+ad,b^(2)c^(2)+a^(2)d^(2)),(1,ca+bd,c^(2)a^(2)+b^(2)d^(2)),(1,ab+cd,a^(2)b^(2)+c^(2)d^(2))|=(a-b)(a-c) (a-d)(b-c)(b-d)(c-d) .

If a sec theta-c tan theta=d and b sec theta+d tan theta=c ,then prove that (c^(2)+d^(2))^(2)-(ac-bd)^(2)=(ad+bc)^(2)

The factors of a ^(2) -b ^(2) - 4c ^(2) + 4d ^(2) - 4 (ad - bc) are :