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enter image description here

I didn't really know how to translate the problem so i just drew a picture. I actually did this and I got 3 but when I checked again I got 4, and now I'm confused. Please let me know if there is something wrong with the problem and I'll clarify

Jean Marie
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pavle
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  • This might help http://geometryatlas.com/entries/43 – gen-ℤ ready to perish Sep 08 '17 at 20:19
  • Find the coordinates of the center of the equitriangle. That is the center of the big circle. That gives you the dimater of big circle. Find the ratio of the circle to the height of the triangle. That will let you find ratio of the triangle to the corner with the small circle. That gives you the ratio of the big circle to the small circle. Good luck. – fleablood Sep 08 '17 at 20:27

2 Answers2

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enter image description here

From this picture it should be clear that the small circles have radiuses $1/3$ of the radius of the big circle.

orangeskid
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Assume side of triangle is 2.

Draw the center of circle. Draw a line from center to vertex of triangle. That bisects the vertex. Draw a line from the center to the perpendicular of the base. That bisects the base.

That creates a 30-60-90 triangle. The base is 1. So the height is $1/\sqrt 3$.

That is the radius of the circle. So the diameter of the circle is $2/\sqrt 3$.

The height of the triangle is $\sqrt 3$. The ratio of diametercircle/heighttriangle is $2/3$.

The corner of the triangle forms a little triangle. Its height is $\sqrt 3 - 2/\sqrt 3=1/\sqrt 3$. So the diameter of the little circle is $\frac 23*\frac 1 {\sqrt3}$.

So the diameter of the big circle is $3$ times bigger than the diameter of the small circle.

So the area of the big circle is $9$ times the area of the small one. And if you add the three small ones, their combined area is 1/3 that of the big circle.

fleablood
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