Recall that the incenter of $\,$$\Delta$$ABC$ $\,$is the center of its incircle and is located at the intersection of the angle bisectors of $\,$$\Delta$$ABC$$\,$.
Morley showed that intersecting adjacent trisectors of $\,$$\Delta$$ABC$$\,$ yields an equilateral triangle (the Morley triangle) inside $\,$$\Delta$$ABC$ $\,$. $\;$Define the Morley center of $\,$$\Delta$$ABC$$\,$$\,$to be the center of its Morley triangle.
The definitions above as well as a little experimentation suggest that both of the following should be true. Are there simple proofs for either?$\;$ Thanks.
Questions: $\,$1) Does the Morley center lie inside the incircle?
$\qquad$$\qquad$ $\,$ 2) Does the incenter lie inside the Morley triangle?