Difference between revisions of "Proofs without words"
m (temp save) |
(→Miscellaneous: enough for now) |
||
Line 137: | Line 137: | ||
== Miscellaneous == | == Miscellaneous == | ||
+ | <center><asy>unitsize(15); defaultpen(linewidth(0.7)); | ||
+ | real a=2.5,b=5,s=a+b; pen colors[] = {rgb(0.9,0.2,0.2), rgb(0.2,0.9,0.2), rgb(0.2,0.2,0.9)}; pen sm = fontsize(8); | ||
+ | void fillrect(pair A, pair B, pen p = invisible, pen l = linewidth(1)){ filldraw(A--(A.x,B.y)--B--(B.x,A.y)--cycle, p, l); } | ||
+ | void htick(pair A, pair B, pair ticklength = (0.2,0)){ | ||
+ | draw(A--B); | ||
+ | draw(A-ticklength--A+ticklength); | ||
+ | draw(B-ticklength--B+ticklength); | ||
+ | } | ||
+ | |||
+ | fillrect((0,0),(s,s)); | ||
+ | fillrect((a,b),(s,s),colors[0]); | ||
+ | filldraw((0,a)--(a,a)--(s/2,s/2)--(a,b)--(a,s)--(0,s)--cycle,colors[1],linewidth(1)); | ||
+ | filldraw((0,0)--(b,0)--(b,b)--(a,a)--(0,a)--cycle,colors[2],linewidth(1)); draw((0,0)--(a,a),linewidth(1)); | ||
+ | |||
+ | draw((s/2,s/2)--(b,a)--(a,a)--(a,b),linewidth(0.7)+linetype("4 2")); | ||
+ | htick((s+1,0),(s+1,b)); | ||
+ | htick((s+1,b),(s+1,s)); | ||
+ | |||
+ | /* in labels, a,b swapped */ | ||
+ | label("$a$",(s+1,b/2),E);label("$b$",(s+1,(s+b)/2),E); | ||
+ | label("$ab$",(a+s,b+s)/2,sm); label("$\frac{(a+b)^2}{4}$",(a,a+s)/2,sm); label("$\frac{a^2}2$",(s/2,a*2/3),sm); label("$\frac{b^2}2$",(a/4,a*2/3),sm); | ||
+ | </asy><br> | ||
+ | |||
+ | The [[Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality|Root-Mean Square-Arithmetic Mean-Geometric Mean]] inequality, <math>\color{red}{ab} \color{black} \le \color{green} \frac{(a+b)^2}{4} \color{black} \le \color{blue} \frac{a^2 + b^2}{2}</math>.<br><br></center> | ||
+ | |||
+ | <center><asy> | ||
+ | unitsize(15); defaultpen(linewidth(0.7)); | ||
+ | void htick(pair A, pair B,pair ticklength = (0,0.15)){ | ||
+ | draw(A--B); | ||
+ | draw(A-ticklength--A+ticklength); | ||
+ | draw(B-ticklength--B+ticklength); | ||
+ | } | ||
+ | real a=10,b=3,r=(a+b)/2; pen sm = fontsize(8), dark = linewidth(1); | ||
+ | pen colors[] = {rgb(0.9,0.2,0.2) + dark, /* GM */ | ||
+ | rgb(0.2,0.9,0.2) + dark, /* AM */ | ||
+ | rgb(0.2,0.2,0.9) + dark, /* QM */ | ||
+ | rgb(0.2,0.9,0.9) + dark }; /* HM */ | ||
+ | pair A = (r-b,(r^2-(r-a)^2)^.5),B=foot((A.x,0),(0,0),A); | ||
+ | |||
+ | draw(arc((0,0),r,0,180)--cycle); dot(A); dot((0,r)); dot((A.x,0)); dot((0,0)); | ||
+ | draw(B--A,colors[3]); label("HM",(A+B)/2, E, sm+colors[3]); | ||
+ | draw((0,0)--(0,r),colors[1]); label("AM",(0,r*2/3), NW, sm+colors[1]); | ||
+ | draw((A.x,0)--A,colors[0]); label("GM",(A.x,A.y/2), SE, sm+colors[0]); | ||
+ | draw((A.x,0)--(0,r),colors[2]); label("RMS",(A.x/5,r*4/5), NE, sm+colors[2]); | ||
+ | |||
+ | draw((-r,0)--A--(r,0), linetype("4 2")); draw((0,0)--B--(A.x,0), linetype("4 2")); draw(rightanglemark((-r,0),A,(r,0))); draw(rightanglemark((0,0),B,(A.x,0))); | ||
+ | |||
+ | htick((-r,-1),(A.x,-1)); htick((A.x,-1),(r,-1)); | ||
+ | label("$a$",((-r+A.x)/2,-1),S); label("$b$",((r+A.x)/2,-1),S); | ||
+ | </asy><!--[[Image:RMS-AM-GM-HM.gif]]--><br> | ||
+ | The [[Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality]].<br><br> | ||
+ | </center> | ||
+ | |||
<center><asy> unitsize(15); defaultpen(linewidth(0.7)); | <center><asy> unitsize(15); defaultpen(linewidth(0.7)); | ||
real r = 0.3, row1 = 3.5, row2 = 0, row3 = -3.5; | real r = 0.3, row1 = 3.5, row2 = 0, row3 = -3.5; |
Revision as of 22:56, 18 March 2010
The following demonstrate proofs of various identities and theorems using pictures, inspired from this gallery.
Contents
[hide]Summations
The sum of the first odd natural numbers is .
The sum of the first positive integers is .
Geometric series
The infinite geometric series .
The infinite geometric series .
Another proof of the identity .
Geometry
Miscellaneous
The Root-Mean Square-Arithmetic Mean-Geometric Mean inequality, .
The Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality.
Fermat's Little Theorem: for (above ).