[Math] Drawing a circle tangential to 3 circles (internally to one of them)

circlesgeometric-constructiongeometry

the image

The two small circles (in black) are equal in radius, and tangential to the large circle. They also touch each other at the center of the large circle.
Now, I want to construct a circle (in orange) which is tangential to the inner two, and the larger circle.
How can I construct it?

Best Answer

I tried it by my own. Hoping there are no mistakes.

Draw a line from the center of the orange circle to the center of the lower small circle. Call the angle from this line the horizontal $\alpha$.

Call the radius of the smaller circle $r$ and the radius of the biggest circle $1$. The radius of the smaller black circles is then obviously $1/2$.

Then you get two equations (from horizontal and vertical distances): $$1=r+(r+1/2)\cos\alpha$$ $$(r+1/2)\sin\alpha=1/2$$ Solving them you get $r=1/3$.

This can be easly constructed:

Draw a horizontal line through the blue center of the big circle and divide it by three. You know how? Then you have the Center of your circle.

[Edit] I used the identity $\cos^2\alpha+\sin^2\alpha=1$. You can skip the step with the angle $\alpha$ and use directly the pythagoras. Which leads to the same.

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