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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