*With the triangle*
angle bisector theorem
and
Morley's trisector theorem
as background ,
are there any pretty theorems known for *triangle* $n$-sectors,
$n > 3$?
For example, angle quadrisectors?

The images below suggest a THEOREM which *I'm hesitant to believe*,
but *illustrates what I seek* :

. Do the $\tfrac{1}{4}$ rays (brown) meet the $\tfrac{1}{2}$ rays (red) as suggested by theQ1black segments, or is that onlyapproximately true?

^{
Quadrisectors. Center: Equilateral triangle. Left & Right: Altitude fixed.
}

^{
Base length: $1$. Altitude: $\sqrt{3}/2$.
}

^{
Left figure enlarged, showing apparent coincidence between
half and quarter angle rays.
}

. Are there any "nice" theorems known for how rays $n$-sectoring the angles of a triangle meet one another?Q2