Mysterious ratio

I need some help or hints with a problem I can't seem to fully understand. I tried to experiment with Geogebra and I think I discovered the solution. My problem is that I don't know the reason why it works. I will try my best to translate the problem.

The problem:

Circle $k$ is given with a point, $K,$ on it and $\lambda$ number. Find all points $M$ where $MT=\lambda \cdot MK$ where $MT$ is tangent to circle $k$ and point $T$ is where the line and the circle meet.

Here is my reconstruction of the problem:

I discovered using Geogebra that the ratio $MT/MK$ is constant if $M$ is on the brown tangent circle that meets the circle $k$ at point $K$. However, I don't know why this occurs and I would like to ask your help to help me understand what's going on.

Thank you!

Solution 1:

Let $H$ be the intersection point of line segment $KM$ with the initial circle $(C)$.

The power of point M with respect to circle $(C)$ can be expressed in two ways:

$$\underbrace{MT^2}_{\lambda^2 MK^2} \ = \ MH \cdot MK$$

Simplifying by $MK$:

$$\lambda^2 MK = MH$$

This relationship between 2 lengths can be given the following vectorialized form:

$$\lambda^2 \vec{KM} \ = \ \vec{HM}$$

(due to the fact that $M,H,K$ are aligned in this order).

$$\lambda^2 \vec{KM} \ = \ -\vec{KH}+\vec{KM}\tag{1}$$

(1) can be given the final form:

$$\vec{KM}\ = \ \underbrace{\frac{1}{1-\lambda^2}}_a \vec{KH}$$

expressing that $H$ has $M$ as its image through a dilation with center $K$ and ratio $a$.

When $H$ describes $(C)$, the locus of its image $M$ is the dilated curve of a circle, i.e., a circle. Moreover, point $K$ being invariant, it is rather easy to establish that this image circle is externally tangent to $(C)$ in $K$.