Thursday, March 17, 2011

Fun with geometry (1)



I found a a couple of fun books of problems in geometry, algebra and probability (geometry book here).

This is one of the problems: given the red circles with radius one-half the large black circle, and the blue circle inscribed so as to just fit inside, derive a relation between the radius of the blue circle and the others. This had me scratching my head for a few minutes before the aha moment.

The challenge question is perhaps easier: prove that the filled-in gray area is equal to the area of one of the red circles.

And a hint for the first problem comes from the next graphic, where I've made a copy of the blue circle and positioned it strategically:


3 comments:

Tobias said...

r(small_circle)=1/sqr(2) * r(big_circle) ?

telliott99 said...

Even simpler. Look at the bottom figure carefully.

Do you like the simple math here?

Tobias said...

i´m such an idiot, can´t use Pythagorean theorem
properly....
now my calculation/proof tells me the same thing than that picture :)

yeah, more simple math here