Take any point x.
Join XA, XB, XC and produce them to points P, Q, R respectively, such that
XP = 3XAXQ = 3XB
XR = 3XCJoin PQ, QR and PR
The sides of the
are three times the corresponding sides of 
are parallel to the sides of
The point X is called the centre of enlargement.
Let us consider various cases for different values of the enlargement factor.
when K > 1
Let K = 2Take O as the centre of enlargement, two possible figures are shown:

(Pre-image)
BC = 4.2 cm
AC = 6 cm
image
QR = 8.4 cm
PR = 12 cm

Enlargement factor less than one.
when K < 1Let K = 1/3
In this case two possible figures are as shown:

(Pre-image)
AB = 4.5 cm
BC = 6 cmAC = 7.5 cm
(Image)
PQ = 1.5 cm
QR = 2.0 cmPR = 2.5 cm

when K = 1

DABC (Pre-image) DPQR (Image)
AB = 2 cm PQ = 2 cmBC = 3 cm QR = 3 cm
AC = 3.5 cm PR = 3.5 cm
To Summarise
(1) In case (iii) DABC is an image of itself.
(2) When K = 1, we get congruent figures and congruent figures are also similar.(3) In each case, image is similar to pre-image.
(4) The ratio of corresponding sides is K.(5) The corresponding sides are parallel to each other.
(6) The ratio of the areas of two similar figures is K2.(7) In solids, the ratio of the volumes of two similar solids will be K3.
