Condition for four points to concyclic

There are 10 points in a plane of which no three points are collinear and four points are concyclic.The number of different circles are can be drawn through at least three points of these points is (A) 116 (B) 120 (C) 117 (D) none of these

Condition for the three points to be collinear Explain by giving an example.

If a line segment joining two points subtends equal angles at two other points lying on the sae side of the line segment; the four points are concyclic.

Theorem: 7 If the line segment joining two points subtends equal angles at two other points lying on the same side of the line segment; the four points are concyclic.i.e lie on the same circle.

Statement -1 : For any four complex numbers z_(1),z_(2),z_(3) and z_(4) , it is given that the four points are concyclic, then |z_(1)| = |z_(2)| = |z_(3)|=|z_(4)| Statement -2 : Modulus of a complex number represents the distance form origin.

Prove that the necessary and sufficient condition for any four points in three- dimensional space to be coplanar is that there exists a liner relation connecting their position vectors such that the algebraic sum of the coefficients (not all zero) in it is zero.

Show that the line segments joining the points (4, 7, 8), (-1, -2, 1) and (2, 3, 4), (1, 2, 5) intersect. Verify whether the four points concyclic.

Write sufficient conditions for a point x=c to be a point of local maximum.

If the curves (x^(2))/(4)+y^(2)=1 and (x^(2))/(a^(2))+y^(2)=1 for a suitable value of a cut on four concyclic points,the equation of a cut on four concyclic through these four points is x^(2)+y^(2)=2 (b) x^(2)+y^(2)=1x^(2)+y^(2)=4(d) none of these