Geometry • Topic 23

Homothety

Homothety (or homothecy) is a transformation that scales a figure relative to a center point. It is a powerful tool for proving that points are collinear or circles are tangent.

Definition

Definition (Homothety)

A homothety centered at

O

with ratio

k

(

k \in R, k \neq = 0

) is a transformation that maps a point

P

P^{'}

such that:

O P^{'} = k OP

Positive $k$ : $P^{'}$ is on the ray $OP$ (Direct Homothety).
Negative $k$ : $P^{'}$ is on the opposite ray from $O$ (Inverse Homothety).

Properties

Collinearity: $P, P^{'}, O$ are always collinear.
Parallelism: A line $ℓ$ maps to a parallel line $ℓ^{'}$ .

Consequence: Angles are preserved (shapes are similar).

3. Circles: A circle maps to another circle.

Two non-concentric circles always have two centers of homothety (Internal and External) that map one to the other.

Practice Problems

Exercise (Problem 1)

Two circles

ω_{1}

and

ω_{2}

are tangent internally at

T

. Prove that a line through

T

cuts the circles at

A

and

B

such that the tangents at

A

and

B

are parallel.

Exercise (Problem 2)

Use homothety to prove that the centroid

G

lies on the Euler line

O H

with

H G = 2 GO

. (Hint: Consider the homothety centered at $G$ with ratio $- 1/2$ mapping $A \to M_{a}$ ).

Exercise (Problem 3)

Given two parallel lines and a point

P

between them, construct a circle tangent to both lines and passing through

P