If `Delta= {:|(0,b-a,c-a),(a-b,0,c-b),(a-c,b-c,0)|`, then show that `Delta` is equal to zero.

If a,b,c are comples number and z= |{:(,0,-b,-c),(,bar(b),0,-a),(,bar(c),bar(a),0):}| then show tha z is purely imaginary

the determinant |{:(a,,b,,aalpha+b),(b,,c,,balpha+c),(aalpha+b,,balpha+c,,0):}|=0 is equal to zero if

If a+b+c ne 0 and |{:(a,b,c),(b,c,a),(c,a,b):}|=0 , then using properties of determinant, prove that a=b=c

If A, B, C are the angle of a triangle and |(sinA,sinB,sinC),(cosA,cosB,cosC),(cos^(3)A,cos^(3)B,cos^(3)C)|=0 , then show that DeltaABC is an isosceles.

If in a DeltaABC, (a cos A+b cos B+ c cos C)/(a sin B+b sin C+ c sin A)=(a+b+c)/(9R) , then prove that Delta is equilateral.

Show that A(a,b,c),B(b,c,a) and C(c,a,b) are the vertices of an equilateral triangle.

If a, b and c are real numbers, and triangle = |[b+c,c+a,a+b],[c+a,a+b,b+c],[a+b,b+c,c+a]| = 0 Show that either a+b+c = 0 or a=b=c

If Delta is the area of a triangle with side lengths a, b, and c, then show that Delta le (1)/(4) sqrt((a + b + c) abc) . Also show that equality occurs in the above inequality if and only if a = b = c

Let hat a , hat b ,and hat c be the non-coplanar unit vectors. The angle between hat b and hat c is alpha , between hat c and hat a is beta and between hat a and hat b is gamma . If A( hat a cosalpha, 0),B( hat bcosbeta, 0) and C( hat c cosgamma, 0), then show that in triangle AB C , (|hat axx(hat bxx hat c)|)/(sinA)=(|hat bxx(hat cxx hat a)|)/(sinB)=(|hat cxx(hat axx hat b)|)/(sinC)

If |[a,b-c,c+b],[a+c,b,c-a],[a-b,a+b,c]|=0 , then the line a x+b y+c=0 posses through the fixed point which is