The following diagram shows an equilateral triangle, a green circle and green semicircle of the same radius and vertically aligned, and a red circle. Wherever things look tangent, they are tangent.
Let's start sliding the red circle and green semicircle together to the right, and see what happens.
Interestingly, the green semicircle touches the right side of the triangle, and the red semicircle touches the green circle, at the same time.
Without calculating any radii, explain why this happens.
Some of the arguments below are the same as those in @DavidG’s answer.
Consider the following figures.
In the left figure, the right edge of the blue triangle is defined as the line that passes through the centre of the green circle and is tangent to the green semicircle. This means, the right blue edge is parallel to the right black edge, so the blue triangle is equilateral.
After sliding until the green semicircle touches the right edge of the big black triangle, i.e., in the right figure, the left edge of the blue triangle passes through the centre of the green circle (because if we ignore all the circles, then the right figure is a reflection of the left figure) and is tangent to the red circle (because it remains tangent throughout the sliding movement).
By the symmetry in the blue triangle, since the red circle touches the green semicircle, it must also touch the green circle.
Because if we add a line parallel to the left edge going through the center of the green circle, we wind up with a smaller equilateral triangle with two green semicircles and a red circle, that is fully symmetric around the angle bisector from the lower left corner. Since the red circle touches the moved green semicircle, it must touch the green circle.
@Pranay asks questions....
An alternate way of looking at this is that we move the red circle, green semicircle, and left triangle edge until the edge goes through the center of the green circle. We can show that the gap from the top of the green circle to the top of the triangle is the same as the radius of the green circle. (Some math on the 30/60/90 triangle from the center, top, and tangent point.) This means that an original line from to center of the green circle and parallel to the left edge is tangent to the green semicircle, and when that line is shifted as well, it is tangent to both the green circle and green semicircle. (We can prove this by shifting the left edge down by the green diameter, so the tangent to the green circle becomes the tangent to the semicircle.) There is also a horizontal line tangent to the green circle and green semicircle (in both positions.) All this is shown here:
At this point, I believe we have enough symmetry to say that the red circle must be touching the green circle and the green semicircle must be touching the right edge.
