Area of Triangles: Moving Partitions

Please see the attached file for the fully formatted problems.

Creating models for incidence axioms.

Construct a model in which axioms I-1, I-2, I-4, and I-5 are valid but I-3 is not. Axioms: I-1 Each 2 distinct points determine a line I-2 Three non-collinear points determine a plane I-3 If 2 points lie in a plane, then any line determined by those 2 points lies in that plane. I-4 If 2 points meet their intersection is a line I-5 Space consists of at least 4 noncoplanar points, and contains 3 noncollinear points. Each plane contains at least three noncollinear points and each line contains at least two distinct points. I can create a five point model where all 5 axioms are valid, but cannot see a model where all but I-3 are valid.

Metric space and triangle inequality.

Prove that in a metric space, if C lies between A and B and O is any other point, then OC<=OA + OB. (Hint make 3 applications of the triangle inequality) Triangle inequality: For triangle ABC AB+BC=>AC

Trigonometric Identity

Verify the following identity: (sin(x) + cos(x))^2 = 1 + sin(2x)

Trigonometric Identity

Verify the following identity: sec(x)-cos(x) = tan(x)sin(x)