Difference between revisions of "User talk:Bobthesmartypants/Sandbox"
(→checkerboasrd) |
(→Fermat point) |
||
Line 445: | Line 445: | ||
==Fermat point== | ==Fermat point== | ||
− | <asy>draw((0,0)--dir(30)*dir(30));</asy> | + | <asy> |
+ | import math; | ||
+ | |||
+ | draw((0,0)--(4,0)--(0,3)--cycle); | ||
+ | draw((0,0)--3*dir(30)--4*dir(-60)--cycle); | ||
+ | draw((4,0)--4*dir(60)--(4*dir(60)+3*dir(30))--cycle); | ||
+ | draw((0,3)--3*dir(150)--(3*dir(150)+4*dir(-60))--cycle); | ||
+ | |||
+ | draw((0,0)--(4*dir(60)+3*dir(30)),blue+linetype("8 8")); | ||
+ | draw((0,3)--4*dir(-60),blue+linetype("8 8")); | ||
+ | draw((4,0)--3*dir(150),blue+linetype("8 8")); | ||
+ | |||
+ | draw((0,0)--(3*dir(150)+4*dir(-60)),red+linetype("8 8 0 8")); | ||
+ | draw((0,3)--4*dir(60),red+linetype("8 8 0 8")); | ||
+ | draw((4,0)--3*dir(30),red+linetype("8 8 0 8"));</asy> |
Revision as of 21:34, 7 June 2014
Contents
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
Picture 2
physics problem
Solution
inscribed triangle
moar images
yay
solution reflection
origami
combos
circles
more circles
checkerboasrd
Fermat point