The blue line segments come from rotating out the two halves of a side of the red triangle about its midpoint, until they are parallel with the other side of the red triangle with which they shared a vertex. The end points of the blue lines then all lie on a Circle. This is true for any triangle.
The centres of pairs of Excircles and two of the end points of two of the blue segments and the vertex between them all lie on a line (green). Each tangent point of an Excircle with a side of the red triangle shares a line with each end point of the blue segment opposite the Excircle, and each of the other tangent points of the same Excircle. Each of these lines each shares a common point of intersection with the bisector of the opposite angle of the red triangle one of the other blue segments, and is perpendicular to the line joining the two tangent points to the extended sides of the red triangle of one of the other Excircles, and the two mentioned points of intersection lie on a line though two of the tangent points to two of the sides of the red triangle and the Incircle of the red triangle. The tangent points to each of two extended sides of the red triangle and an Excircle lie on a line through the end points of two of the blue segments that are inside the Excircle.
Music: "Mysterious Kyoto", Japanese Traditional Music (No Copyright).
• Japanese traditional m...