Given a cube ABCD A_(1)B_(1)C_(1)D_(1) which lower base ABCD, upper base A_(1)B_(1)C_(1)D_(1) an...

Given a cube ABCDA_1 B_1C_1D_1 with lower base ABCD and the upper baseA_1B_1C_1D_1 and the lateral edges A A_1, BB_1,, C C_1, DD_1. M and M_1, are the centres of the faces ABCD, and A_1 B_1 C_1 D_1 respectively. O is a point on the line MM_1. Such that overline(OA) + overline(OB) + overline(OC) + overline(OD) = overline(OM_1), . if overline(OM) = lambda overline(OM_1), then lambda is equal to

Let A_(1),B_(1).C_(1).D_(1) be the midpoints of the sides of a convex quadrilateral ABCD and let A_(2).B_(2),C_(2),D_(2) be the midpoints of the sides of the quadrilateral A_(1)B_(1)C_(1)D_(1). If A_(2),B_(2),C_(2),D_(2), is a rectangle with sides 4 and 6, then what is the product of the lengths of the diagonals of ABCD?

Let L_(1) and L_(2) be two lines intersecting at P If A_(1),B_(1),C_(1) are points on L_(1),A_(2),B_(2),C_(2),D_(2),E_(2) are points on L_(2) and if none of these coincides with P, then the number of triangles formed by these 8 points is