![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi09vvpslrdr5REUBI2Xqnd1t8NDaIw4nb1PMMmFgVHL-0cdEKFuZD5QvYZKw_ZDmdvb1pE6wwvmeo-Nou2TR-wGxnuWvDbWP-q-PWNgR4UyKzkFWQxCHx5lUPMm0m6Z4gGAhprms74vco3/s320/Screen+shot+2011-03-17+at+8.36.34+AM.png)
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:
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg04O1lwl4t-0wtBGwfBMnnvCgi1Sj5UavtAEQlQFlem5CCekwMtcPqRMiFfGeSOfpEwrT-gkSewVy7UAmKJcY1ZaTcW8YF6F4bE0G8tNR-OBySGsRxH8NexpH6ZPi0BzjFpjqq9t-6cNdb/s320/Screen+shot+2011-03-17+at+8.36.49+AM.png)