**Similar Figures:**

*Definition:*

Polygons which are equiangular and have their corresponding sides proportional are called similar.

If, in addition, their corresponding sides are parallel, they are said to be similarly situated or homothetic.

**Theorem 1:**

If O is any fixed point and ABCD…X any polygon, and if points are taken on OA, OB, OC, …OX (or those lines produced either way) such that , then the polygons and are homothetic.

Before we state the next theorem, some background is necessary.

If O is a fixed point and P is a variable point on a fixed curve S, and if is a point on OP such that , a constant, then the locus of is a curve , which is said to be homothetic to S; and are corresponding points.

O is called the centre of similitude of the two figures.

If and lie on the same side of O, the figures are said to be directly homothetic w.r.t.O, and O is called the external centre of similitude.

If and lie on the opposite sides of O, the figures are said to be inversely homothetic w.r.t. O, and O is called the internal centre of similitude.

If we join A and B in the first case, we say that the parallel lines are drawn in the same sense, and in the second case, in opposite senses.

**Theorem 2:**

Let A, B be the centres of any two circles of radii a, b; AB is divided externally at O and internally at in the ratio of the radii, that is, then the circles are directly homothetic w.r.t. O and inversely homothetic w.r.t. , and corresponding points lie on the extremities of parallel radii.

**Now, prove the above two hard core basic geometry facts. ! π π πΒ **

Cheers,

Nalin Pithwa.