- 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
First, continue to hit at . Also continue to hit at .
We have that . Because , we have .
Similarly, because , we have .
We also have that because is a parallelogram, and .
Therefore, . This means that , so .
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 .
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.
Cyclic squares DOTS DTOS TDORS