Moving_Similar

Basic theorem for moving polygons

Consider a polygon (for simplicity a triangle) DEZ, moving so that a vertex D remains fixed, another vertex E moves on a circle, while the polygon as a whole remains similar to itself (here remains always similar to ABC). Then the other vertices (here Z) move on fixed circles

Construct (DHI) similar to (ABC), where H is the center of the given circle, where the vertex E moves. I is a fixed point and the triangles (HDE), (IDZ) are similar. Their similarity ratio k = HD/DI = AB/AC is known, hence the length of IZ is fixed and Z moves on the circle with center I and radius (IZ).

A similar result holds when E moves on a line, while D remains fixed and the polygon remains similar to itself. Then every other vertex of the polygon describes a fixed line. For an application of this result to the geometry of the triangle look at TriangleBisectors.html .

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