The diagram shows an equilateral triangle, a black circle and black semicircle of the same radius and vertically aligned, a green circle and a red circle. Wherever things look tangent, they are tangent.
Which is bigger, the green circle or the red circle?
The
red
is bigger.
Picture:
Explanation:
The red and green circles can be mirrored across the height of the triangle (orange in middle panel) into being tangent to the same "scaffold" consisting of the half shown black circle and one of the slanted sides (let's pick the left WLOG) of the triangle. The question therefore becomes: Relative to the axis of symmetry of this scaffold (dashed blue line in last panel) which circle is farther out?
This is slightly complicated by the fact that they lie on opposite sides of that axis of symmetry.
Observe that the black circles' radius is a quarter of the triangle's height. (This follows, for example, from the fact that the circumradius of an equilateral triangle is twice the inradius.) It follows that the line connecting the midpoints of the left and bottom sides (cyan) is tangent to the upper black circle. Conveniently, this line is the mirror image of the base line w.r.t. scaffold symmetry. From this we can already conclude that the red circle is no smaller than the green because the green lies inside the small triangle (it touches the base from the inside) whereas the red touches the upper black circle which lies on the outside of the cyan line. Unless the red circle and cyan line touch the black circle at the same point the red circle intersects the cyan line and is therefore farther out than the green circle. The red and black circles and the cyan line would touch in a single point if and only if the black and red circles are the same size which is not the case.
I have accepted @Albert.Lang's answer.
Here is another way to see it.
Explanation:
Reflect the red circle into the pink circle of the same size. The presence of the orange line forces the green circle to be further up the central black circle than the pink circle (and both are constrained by the top horizontal line), so the green circle must be smaller than the pink/red circle.
But what if...
...the point of tangency between the green circle and the orange line, happens to coincide with the point of tangency between the left black circle and the orange line? In that case, the previous argument would not work. But -
That doesn't happen:
In case you're wondering, the ratio of the green circle's radius to the red circle's radius is exactly:
$\dfrac{10\sqrt2-6\sqrt3-4\sqrt6+15}{9}\approx0.99465$
Whoever downgraded my 1st answer, please explain why:
