User:Gavintlgold~enwiki

Gavin is a 11th grader of Green Meadow Waldorf School (The Wikipedia entry for the school is almost entirely edited by the math teacher, username hgilbert, by the way). He is interested in computers and has built his own PC running Ubuntu.

He is currently excited about the fact that Neil J. Patel is coming out with a media center for linux called Arena.

In the old days, his blog was named "The Putta Butta." But, alas, after many searches (using Wikipedia, of course), he has discovered an embarrasment: his blog title means 'whore pig'. He has therefore changed the name to a more responsible title, namely "The Non-Offensive Blog Title." The legend lives on, however, in the URL of the blog, and in the blog's archives.

That aforementioned math teacher says that he has discovered a new principle of geometry, namely the way to prove that a rhombus is half its diagonal times its horizontal. He calls it "The Gavin Theorem." Gavin himself assures all that ask that this is surely already a known fact. Sadly, Gavin does not want to reveal the secrets of the Theorem for fear that its knowledge will lead to the eventual destruction of the world.

The Theorem is NOT this though:

Let A, B, C and D be the vertices of the rhombus, named in agreement with the figure (higher on this page). Using $$\overrightarrow{AB}$$ to represent the vector from A to B, one notices that $$\overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC}$$ $$\overrightarrow{BD} = \overrightarrow{BC}+ \overrightarrow{CD}= \overrightarrow{BC}- \overrightarrow{AB}$$. The last equality comes from the parallelism of CD and AB. Taking the inner product,

<\overrightarrow{AC}, \overrightarrow{BD}> = <\overrightarrow{AB} + \overrightarrow{BC}, \overrightarrow{BC} - \overrightarrow{AB}>$$
 * $$= <\overrightarrow{AB}, \overrightarrow{BC}> - <\overrightarrow{AB}, \overrightarrow{AB}> + <\overrightarrow{BC}, \overrightarrow{BC}> - <\overrightarrow{BC}, \overrightarrow{AB}>$$
 * $$ = 0$$

since the norms of AB and BC are equal and since the inner product is bilinear and symmetric. The inner product of the diagonals is zero if and only if they are perpendicular.