# User talk:Bobthesmartypants/Sandbox

## Contents

- 1 Bobthesmartypants's Sandbox
- 2 Solution 1
- 3 Solution 2
- 4 Picture 1
- 5 Picture 2
- 6 physics problem
- 7 Solution
- 8 inscribed triangle
- 9 moar images
- 10 yay
- 11 solution reflection
- 12 origami
- 13 combos
- 14 circles
- 15 more circles
- 16 checkerboasrd
- 17 Fermat point
- 18 cenn driagrma
- 19 cyclic square
- 20 diagram
- 21 Cyclic squares DOTS DTOS TDORS

## Bobthesmartypants's Sandbox

## Solution 1

First, continue to hit at . Also continue to hit at .

We have that . Because , we have .

Similarly, because , we have .

Therefore, .

We also have that because is a parallelogram, and .

Therefore, . This means that , so .

Therefore, .

## Solution 2

Note that is rational and is not divisible by nor because .

This means the decimal representation of is a repeating decimal.

Let us set as the block that repeats in the repeating decimal: .

( written without the overline used to signify one number so won't confuse with notation for repeating decimal)

The fractional representation of this repeating decimal would be .

Taking the reciprocal of both sides you get .

Multiplying both sides by gives .

Since we divide on both sides of the equation to get .

Because is not divisible by (therefore ) since and is prime, it follows that .

## Picture 1

Two half-circles are drawn as shown above, with a line throught the two intersections points, of the half-circles. Lines for parallel to the bases of the half-circles are drawn such that the distances between and and and are always the same for all .

The intersection points of with one of the half-circles are labeled , and with the other half-circle at , as shown in the diagram.

Prove that

## Picture 2

## physics problem

## Solution

## inscribed triangle

## moar images

## yay

## solution reflection

## origami

## combos

## circles

## more circles

## checkerboasrd

## Fermat point

## cenn driagrma

## cyclic square

## diagram

## Cyclic squares DOTS DTOS TDORS