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:


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
now my calculation/proof tells me the same thing than that picture :)

yeah, more simple math here

epittelkau said...

One cannot assume that three blue circles fit across the diameter of the largest circle. They do, but one can't make that assumption. However, the problem is easily solved using Descartes Theorem.

telliott99 said...

Like I said, it's a hint! The point is to show the way to the proof, which is very easy when you see it. Thanks for the mention of Descartes Theorem, which I didn't know about.