Small Geometry Puzzle

[ch3cooh]

Challenge: write up your initial instinct, then try to prove it and let me know what happens. :D
I really like this puzzle.  And I'm sure that there's a simpler proof than the one I'm using right now.

Puzzle:



Solution:
The orange is exactly the same area as the sum of the two blue areas.

Proof:
Step 1: The Orange circle is exactly 1/4 of the big circle.  Proof by picture:

(Draw another equlateral triangle inscribed in the circle, with a 180 degree rotation relative to the original triangle.  These two triangles overlap in a regular hexagon that can be split into 6 triangles like a pie.  Each of these triangles is the same size as any of the 6 points of the star now inscribed in the circle.  So it is clear that the radius of the orange circle is half the radius of the large circle.  Since A = pi*r^2, this means that the area of the orange circle is 1/4 the area of the whole circle.

Step 2: Blue area is exactly 1/3 of the remaining area in the large circle that is not part of the orange circle.  Proof by picture:

The remaining area has three semetrical components.

Step 3: Since the orange circle is 1/4 of the large circle, the remaining interior of the large circle is 3/4 of the area.  The blue area is 1/3 of that, and 1/3 of 3/4 is 1/4.  So both the orange region and the combined two blue regions are 1/4 the area of the whole circle.
Boom!  Done!

Using this to draw a very pretty infinite summation picture. :)

1/4 + 1/16 + 1/64 + ... 1/4^n  as n approaches infinity = 1/3

Comments

[ch3cooh.livejournal.com]

I've been in a fairly Euclidean-geometry-mindset saturated state lately, so the potential of triangles and hexagons really pop out at me. Squares and rectangles on the other hand... these are tricky.

In the site I linked to below, I haven't yet figured out how to construct a square in 8 moves, and I've tried at it for more than an hour ;-P Also, for the proof above, as I mentioned, there's probably an easier way to see the 1/4 relationship than the star I thought of, but I don't know it. Who knows, maybe squares are involved somehow. :)

Construction game:
http://sciencevsmagic.net/geo/
Or, this is a simpler but less 'visually clean' version:
http://euclidthegame.com/

Squares are so hard! My TA told me today that there's a compass & straight edge method for, given a rectangle of any area (even one with an irrational area) constructing a square that has the same area. Still trying to figure it out...

[cesium12.livejournal.com]

Hm, I think I see how to do that. rot13 for spoilers:

Gb pbafgehpg gur fvqr bs lbhe fdhner, lbh arrq gur trbzrgevp zrna bs gur fvqrf bs lbhe erpgnatyr.
Va nal pvepyr, vs pubeqf no naq pq vagrefrpg ng r, gura nr gvzrf ro rdhnyf pr gvzrf rq. V'z fher gurer'f n anzr sbe guvf cebcregl ohg V qba'g erzrzore vg.
Gurersber, sbe lbhe erpgnatyr jvgu fvqrf k naq l, pbafgehpg gur pvepyr jvgu qvnzrgre k cyhf l, naq qenj n crecraqvphyne pubeq guebhtu vg yvxr n puevfgvna pebff. Gur arj pubeq vf qvivqrq vagb gjb rdhny frtzragf bs yratgu fdeg bs k gvzrf l. Qbar!