More Geometry

(See attached file for full problem description)

More & More Geometry

(See attached file for full problem description)

Proofs

Prove that a triangle is equilateral <=> its incenter equals its centroid.

Differentiable Geometry, diff structure for manifold.

Differentiable Geometry. Diff structure for manifold. -------------------------------------------------------------------------------- Let S^1 a subset of R^2 be the unit-circle; that is, S^1 = { (x,y) element of R^2 : x^2 + y^2 = 1}. (a) Explicitly find a differentiable structure for S^1 ( and indicate what the dimension of the manifold S^1 is with your differentiable structure). (b) Specifically describe how you will go about making your structure above maximal.

proves

1- prove that if AF/FB = AF'/F'B where A, B, F, F' are collineasr and distinct then F does not have to equal F' 2-suppose that the sides AB, BC, CD and DA of a quadrilateral ABCD are cut by a line at the points A' B' C' D' respectively, show that AA'/A'B * BB'/B'C * CC'/C'D * DD'/D'A = +1