LOGIC to prove SIMILAR Objects are EQUAL

While developing GEM (GEstures with Mouse) the biggest problem i faced is, "the user drawn gesture is most of the times smaller / bigger than already defined gesture" but they are almost same in shape. So, i used a logic to prove SIMILAR Objects (Different in size but with same shapes ) are EQUAL.

LOGIC:
Let us consider two similar circles A, B as shown in picture.

The co-ordinates of Circle A are
(x1, y1) = (6, 4)
(x2, y2) = (8, 6)
(x3, y3) = (6, 8)
(x4, y4) = (4, 6)

The co-ordinates of Circle B are
(X1, Y1) = (6, 3)
(X2, Y2) = (9, 6)
(X3, Y3) = (6, 9)
(X4, Y4) = (3, 6)

now we re-calculate the  above co-ordinates using following formulas

X =  [X - [Xmax+Xmin]/2] / [Xmax-Xmin]+ 0.5 ---------(1)

        where Xmin = min(x1,x2,x3,x4) for circle A
                   Xmax = max(x1,x2,x3,x4) for circle A
                   X = x1, x2, x3, x4 for circle A

                   Xmin = min(X1,X2,X3,X4) for circle B               
                   Xmax = max(X1,X2,X3,X4) for circle B
                   X = X1, X2, X3, X4 for circle B

Y =  [Y - [Ymax+Ymin]/2] / [Ymax-Ymin]+ 0.5 ------------(2)

        where Ymin = min(y1,y2,y3,y4) for circle A
                    Ymax = max(y1,y2,y3,y4) for circle A
                    Y = y1, y2, y3, y4 for circle A
                   
                   Ymin = min(Y1,Y2,Y3,Y4) for circle B
                   Ymax = max(Y1,Y2,Y3,Y4) for circle B
                   Y = Y1, Y2, Y3, Y4 for circle B
                  
Re-Calculating Circle A points:

Xmin = min(6, 8, 4, 6) = 4
Xmax = max(6, 8, 4, 6) = 8


Ymin = min(4, 6, 6, 8) = 4
Ymax = max(4, 6, 6, 8) = 8

new x1 =  (6 - (8+4) / 2)/4 + 0.5 = 0.5
new y1 =  (4 - (8+4) / 2)/4 + 0.5 = 0.0

in the same manner

new x2 = 1.0
new y2 = 0.5

new x3 = 0.5
new y3 = 1.0

new x4 = 0.0
new y4 = 0.5

Re-Calculating Circle B points:

Xmin = min(6, 9, 6, 3) =3
Xmax = max(6, 9, 6, 3) =9

Ymin = min(3, 6, 9, 6) =3
Ymax = max(3, 6, 9, 6) =9

new X1 = (6 - (9+3)/2)/(9-3) + 0.5 = 0.5
new Y1 = (3 - (9+3)/2)/(9-3) + 0.5 = 0.0

in the same manner

new X2 = 1.0
new Y2 = 0.5

new X3 = 0.5
new Y3 = 1.0

new X4 = 0.0
new Y4 = 0.5

After Re-calculating Circle A & B points using Formula (1) and (2)

x1 = X1; x2 = X2; x3 = X3; x4 = X4; and 
y1 = Y1; y2 = Y2; y3 = Y3; y4 = Y4;


Hence, SIMILAR Circles A and B are EQUAL

Note: Not only circles using this formulas any two similar drawing shapes can be proved to equal.