1.

Points A(veca), B(vecb), C(vecc) and D(vecd) are related as xveca+yvecb+zvecc+wvecd=0 and x+y+z+w=0, where x, y, z and w are scalars (sum of any two of x, y, z and w is not zero). Prove that if A, B, C and D are concyclic, then |xy||veca-vecb|^(2)=|wz||vecc-vecd|^(2).

Answer»

Solution :From the given conditions, it is CLEAR that POINTS `A(veca), B(vecb), C(VECC) and D(vecd)`are coplanar.
Now, `A, B, C and D` are concyclic. THEREFORE,
`""APxxBP = CPxxDP`
`""|(y)/(x+y)||veca-vecb||(x)/(x+y)||veca-vecb|=|(w)/(w+Z)||vecc-vecd||(z)/(w+z)||vecc-vecd|`
`""|xy||veca-vecb|^(2)= |wz| |vecc-vecd|^(2)`


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