The medial triangle of a triangle ABC is the triangle whose vertices are located at the midpoints of the sides AB, AC, and BC of triangle ABC. From an arbitrary point O that is not a vertex of triangle ABC, you may take it as a given fact that the location of the centroid of triangle ABC is the vector (vector OA + vector OB + ve

1. Write a proof. AB*BE = CB*BD. Prove triangle ABC is congruent to triangle DBE.
2. A parent group wants to double the area of a playground. The measurements of the playground are width is 2W and the length is 2L. They ask you to comment. What would you say?
3. Find the length of the altitude drawn to the hypotenuse.

Prove the CENTROID therorem using the VECTOR proof as well as the SYNTHETIC proof
Explain how to derive the distance formula (assuming that the distance formula is not yet known), first in 2 dimentional and then in 3 dimentional

Write a program that reads three real numbers, assigns the appropriate boolean value to the following boolean variables, and displays the associated values.
Triangel: true if the real numbers can represent lengths of the sides of a triangle (the sum of any two of the numbrs must be greater than the third); false otherwise.

3. This exercise is about the inclusion-exclusion principle.
a) Let X and Y be finite ts and suppose that |X| = 11, |Y| = 6, and
|X∩Y| =4. Find |XUY|.
b) Suppose that U is a finite universal set. If |U| = 21, |XUY| = 11. |X| = 4 and |Y|= 10. find |XcUYc|.
c) Each tile in a collection of 19 is a square or a triangle and

Task:
A. Discuss differences between neutral geometry and Euclidean geometry.
B. Explain the importance of Euclid's parallel postulate and how this was important to the development of hyperbolic and spherical geometries.
Note: Euclid's parallel postulate states the following: "For every line l and for every external