How to calculate the area of the rectangle | 3 Circles insides a rectangle | Tricky Math Question

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[GoogleAccount-pict]

Pretty common trick, but pretty cool. Can I suggest covering AIME II 2023 Problem 13 and USAMO 2010 Problem 1?

[glizio]

I chose to take the altitude of the equilateral triangle created after connecting the 3 radii. 30-60-90 so it’s rad3 and then from there it’s just 2+ rad3 as the height and 4 length. 4(2 + rad3) = 8 + 4rad3

[SOURAVKUMAR-ttbx]

But sir i have a doubt how can we say that that the vertical line will be tangent to the bottom circle i know it seems pretty obvious by looking st the symmetry but can you still tell me the mathmatical reason that why the vertical line is tangent to the bottom circle