under certain direct similarities, more precisely certain rotations the center of which is the center of the polygon, and a composition of direct similarities is also a direct similarity. For example we see the image of the initial regular pentagon under a homothety of negative which is a similarity of ±180° angle and a positive ratio Below the title on the right, the second image shows a similarity decomposed into a rotation and a homothety. Similarity and rotation have the same angle of +135 degrees modulo 360 degrees. Similarity and homothety have the same ratio multiplicative inverse of the (square root of 2) of the inverse similarity. Point S is the common center of the three transformations: rotation, homothety and similarity. For example point W is the image of F under the rotation, and point T is the image of W under the homothety, more briefly T = (W ) = ( F )) = ( F ) = ( F ), by naming the previous rotation, homothety and similarity, with This direct similarity that transforms triangle EFA into triangle ATB can be decomposed into a rotation and a homothety of same center S in several manners. For example, the last decomposition being only represented on the image. To get we can also compose in any order a rotation angle and a homothety With and if is the reflection with respect to line (CW ), then is the indirect similarity that transforms segment [BF ] into segment [CT ], but transforms point E into B and point A into A itself. Square ACBT is the image of ABEF under similarity Point A is the center of this similarity because any point K being invariant under it fulfills only possible otherwise written How to construct the center S of direct similarity how to find point S center of a rotation of +135° angle that transforms ray [SE ) into ray [SA )? This is an inscribed angle problem plus a question of orientation. The set of points is an arc of circle that joins E and A, of which the two radius leading to E and A form a central angle This set of points is the blue quarter of circle of center F inside square ABEF. In the same manner, point S is a member of the blue quarter of circle of center T inside square BCAT. So point S