Does the similar figures theorem apply also to altitudes and medians?

A similar figure is a figure that is geometrically the same.

In the case of triangles, two triangles are similar if their sides are proportionate.

I assume we are talking about the theorem of scaling.

If we have a triangle and we scale by a factor `a/b`, all sides of the scaled triangle will be increased by `a/b`.

Since a triangle is uniquely defined by the length of its sides, all angles will remain the same and consequently the altitude will increase by `a/b`

The medians will still be in the same relative position, by definition of a median. The length from a vertex to the opposite side will increase by the factor `a/b`. ( this is a linear measurement )

The easiest way to see this is:

All linear measurements i.e. length will increase by `a/b`, all quadratic measurements will increase by `(a/b)^2` and all cubic measurements will increase by `(a/b)^3`. Angular measurement remain unchanged by scaling, these are non linear, non quadratic and non cubic.

As an example:

Given an equilateral triangle with sides 4.

Scaled by a factor `1/2`

Altitude of triangle:

`sqrt(4^2-2^2)=sqrt(12)=2sqrt(3)`

After scaling by `1/2`:

`"sides"=2`

Altitude of scaled triangle:

`sqrt(2^2-1^2)=sqrt(3)`

Notice this is:

`1/2xx2sqrt(3)=sqrt(3)`

Notice area:

Area of original triangle:

`1/2(4)xx2sqrt(3)=4sqrt(3)`

Area of scaled triangle:

`1/2(2)xxsqrt(3)=sqrt(3)`

`(1/2)^2xx4sqrt(3)=1/4xx4sqrt(3)=sqrt(3)`

As you can see the area is quadratic, so we used `(a/b)^2`

Don't know whether this helps you.