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 ...continues

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

proves

1- the incircle of triangle ABC touches the sides BC, CA, AB at the points x,y,z respectively. yz is produced to meet BC at k. show that Bx/Cx = Bk/Ck

Angles/Pounds

A right triangle has one angle that measures 90 degree. If one of the acute angles is 2 times the size of the other, what is the measure of the smallest angle? Explain your answer. The cake receipe calls for 14 ounces of cream cheese. How much of the cream cheese will be left over? Show your answer in pounds and ounces.

Differential geometry/differentiable manifolds

Let M be a differentiable manifold such that M is orientable and connected. Prove that there exist exactly two distinct orientations on M.

Differentiable manifolds

Let M and N be differentiable manifolds. Let alpha: M -> N be a local diffeomorphism. Prove that if N is orientable, then M is orientable.

Proofs

1- prove that in a cyclic quadrilateral we have opposite angles supplementary without introducing the center of the circle. 2- prove that a quadrilateral is paralleogram if it is simple and opposite angles are congurent.

Circular Table

A circular table measures about .9 m around its edge. What is the approximate length of its diameter. Would you please explain this to us and how to work the problem. Thank you.