Cabri Java Applet
Centroid of 10-gon

Drag a vertex of the 10-gon. The geometrical center(centroid) will move along with it.
This allpet shows how you can construct the center of a 10-gon step by step.

Let's start with a triangle.
The intersection of two medians is the centroid of a triangle.
You can create a macro(C3) by defining three vertices as the initial objects and the centroid of a triangle as the final object.

Next let's construct the centroid of a quadrilateral, or a tetragon(4-gon).
A quadrilateral consists of two triangles.
Construct a centroid of each triangle using macro(C3). The centroid of the quadrilataral locates somewhere on the line(l1) connecting the two centroids.
The same quadrilateral can be divided into another pair of triangles.
Construct a centroid of each triangle using macro(C3). The centroid of the quadrilataral also locates somewhere on the line(l2) connecting the two centroids.
You find the centroid of the quadrilateral on the intersection of these two lines (l1 and l2).
Thus you can create a new macro(C4) by defining four vertices as the initial objects and the centroid of a quadrilateral(4-gon) as the final object.

Next let's construct the centroid of a hexagon, or 6-gon.
You might have been aware of your task.
A hexagon(6-gon) consists of two quadrilaterals(4-gon).
Construct a centroid of each 4-gon using macro(C4). The centroid of the 6-gon locates somewhere on the line(l1) connecting the two centroids.
The same 6-gon can be divided into another pair of 4-gons.
Construct a centroid of each 4-gon using macro(C4). The centroid of the 6-gon also locates somewhere on the line(l2) connecting the two centroids.
You find the centroid of the 6-gon on the intersection of these two lines (l1 and l2).
Thus you can create a new macro(C6) by defining six vertices as the initial objects and the centroid of a 6-gon as the final object.

Finally you are going to construct the centroid of a decagon, or 10-gon.
You must know your task now.
Yes, a decagon(10-gon) consists of two hexagons(6-gon).
Construct a centroid of each 6-gon using macro(C6). The centroid of the 10-gon locates somewhere on the line(l1) connecting the two centroids.
The same 10-gon can be divided into another pair of 6-gons.
Construct a centroid of each 6-gon using macro(C6). The centroid of the 10-gon also locates somewhere on the line(l2) connecting the two centroids.
You find the centroid of the 10-gon on the intersection of these two lines (l1 and l2).
Thus you can create a new macro(C10) by defining ten vertices as the initial objects and the centroid of a 10-gon as the final object.
As multiple macroes are allplied, the dragging movement may be very slow at first.
Once it moves,the dragging will be smooth.