For any triangle let's construct an orthocenter(H) and centroid(G).
We can consider a line passing through H and G and expect that circumcenter is on the line.
If we define the intersect of the line and the perprndicular bisector of side BC as X,
Since both line AH and line XD are perpendicular of side BC, they are parallel each other and by the theorem of parallel lines,
Also, since G is centroid, length of AG = 2 length of GD
We have to show that X is circumcenter (i.e.the point Y which the perpendicular bisector of side AC intersect with the line HG is exactly to be X).
Likewise the discussion above,
By (*) and (**), Y must be same with X. Therefore X is a circumcenter.
Let's try to make sure by using measure in GSP for many cases.
Case 1. Acute triangle
Case 2. Right triangle
The point which is on right angle becomes a orthocenter and the midpoint of hypotenus becomes a cicumcenter.Therefore by the property of centroid HG=2GX
Case 3. Obtuse triangle
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