`((b^3c^(-2))/(b^(-4)c^3))^(-3)-:((b^(-1)c)/(b^2c^(-2)))^5=b^x c^y` prove that `x+y+6=0`

Let P(x)=((x-a)(x-b))/((c-a)(c-b))c^(2)+((x-b)(x-c))/((a-b)(a-c))a^(2)+((x-c)(x-a))/((b-c)(b-a))b^(2) Prove that P(x) has the property that P(y)=y^(2) for all y in R .

Let P(x) equiv ((x-a)(x-b))/((c-a)(c-b)).c^2+((x-b)(x-c))/((a-b)(a-c)).a^2+((x-c)(x-a))/((b-c)(b-a)).b^2 Prove that P(x) has the properly that P(y) = y^2 for all y in R.

If a ,\ b ,\ c >0\ a n d\ x ,\ y ,\ z in R , then the determinant |\ \ (a^x+a^x)^2(a^x-a^(-x))^2 1(b^y+b^(-y))^2(b^y-b^(-y))^2 1(c^z+c^(-z))^2(c^z-c^(-z))^2 1| is equal to- a. a^x b^y c^x b. a^(-x)b^(-y)c^(-z)\ c. a^(2x)b^(2y)c^(2x) d. zero

if (x_(1),x_(2))^(2)+(y_(1)-y_(2))^(2)=a^(2), (x_(2)-x_(3))^(2)+(y_(2)-y_(3))^(2)=b^(2) (x_(3)-x_(1))^(2)+(y_(3)-y_(1))^(2)=c^(2). where a,b,c are positive then prove that 4 |{:(x_(1),,y_(1),,1),(x_(2) ,,y_(2),,1),( x_(3),, y_(3),,1):}| = (a+b+c) (b+c-a) (c+a-b)(a+b-c)

if (x_(1)-x_(2))^(2)+(y_(1)-y_(2))^(2)=a^(2), (x_(2)-x_(3))^(2)+(y_(2)-y_(3))^(2)=b^(2) (x_(3)-x_(1))^(2)+(y_(3)-y_(1))^(2)=c^(2). where a,b,c are positive then prove that 4 |{:(x_(1),,y_(1),,1),(x_(2) ,,y_(2),,1),( x_(3),, y_(3),,1):}| = (a+b+c) (b+c-a) (c+a-b)(a+b-c)

if (x_(1),x_(2))^(2)+(y_(1)-y_(2))^(2)=a^(2), (x_(2)-x_(3))^(2)+(y_(2)-y_(3))^(2)=b^(2) (x_(3)-x_(1))^(2)+(y_(3)-y_(1))^(2)=c^(2). where a,b,c are positive then prove that 4 |{:(x_(1),,y_(1),,1),(x_(2) ,,y_(2),,1),( x_(3),, y_(3),,1):}| = (a+b+c) (b+c-a) (c+a-b)(a+b-c)

If (x_1-x_2)^2+(y_1-y_2)^2=a^2,(x_2-x_3)^2+(y_2-y_3)^2=b^2 and (x_3-x_1)^2+(y_3-y_1)^2=c^2 then prove that 4|{:(x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1):}|= (a+b+c)(b+c-a)(c+a-b)(a+b-c)