Mass Points Geometry

Some of the harder/hardest questions at Mathcounts can be tackled at ease using mass point geometry
so spend some time understanding it.

Basics 

2014-15 Mathcounts handbook Mass Point Geometry Stretch
from page 39 to page 40

(Talking about motivation, yes, there are students already almost finish
this year's Mathcounts' handbook harder problems.)

From Wikipedia

From AoPS

Mass Point Geometry by Tom Rike

Another useful notes 

Videos on Mass Point :

Mass Points Geometry Part I 

Mass Points Geometry : Split Masses Part II 

Mass Points Geometry : Part III 

other videos from Youtube on Mass Points

It's much more important to fully understand how it works, the easier questions the weights align
very nicely.

The harder problems the weights are messier, not aligning nicely, so you need to find ways to may them integers (LCM) for easier solving.

Let me know if you have questions. I love to help (:D) if you've tried.

Face Diagonal and Space Diagonal of a Rectangular Prism

Face diagonal and space diagonal of a cube 

Ways to calculate face and space diagonal.

Each side of the cube is x units long.

Use  45-45-90 degree angle ratio
( 1 - 1 - √ 2  ) or Pythagorean theorem to get the face diagonal.

Using Pythagorean theorem twice and you'll get the space diagonal.







Face diagonal and Space diagonal of a rectangular prism.

Same way to figure out the face

diagonal of a rectangle as well as

space diagonal of a rectangular prism.

Use Pythagorean theorem or

30-60-90 degree angle ratio

(1 -√ 3  - 2) to figure out the face

diagonal and Pythagorean theorem

twice to figure out the space diagonal.