This note is a compilation of calculations needed in the solutions to the problems titled Dynamic Geometry: P32, 77, 89, 94, and 98 so far by @Valentin Duringer.
Centers and Radii
Let the center of the large semicircle be , the origin of the -plane, its radius , if it is not given, and its diameter . Let the center and radius of the cyan semicircle be and and those of the green semicircle be and . Then , , and .
Let the center and radius of the left yellow circle be and , those of the right yellow circle be and , be the vertical dividing red segment, be perpendicular to , and be perpendicular to . Then by Pythagorean theorem,
Since if we flip the yellow circles and the semicircles below them horizontally about , becomes and becomes . We get the equation for by swapping and . Then . Similarly, . And .
Vertices of Triangles
Let , , and be the points where the left yellow circle tangent to the semicircle below it, the segment , and the unit semicircle respectively. Let the corresponding points of the right yellow circle be , , and respectively.
Now let be perpendicular to . Then
Again we can find the coordinates of , , and by swapping and . So in summary: