Question

1. Does the centroid always lie inside the triangle?

Answer 1

The centroid is always inside the triangle
Each median divides the triangle into two smaller triangles of equal area.
The centroid is exactly two-thirds the way along each median.
Put another way, the centroid divides each median into two segments whose lengths are in the ratio 2:1, with the longest one nearest the vertex. These lengths are shown on the one of the medians in the figure at the top of the page so you can verify this property for yourself.