If O, H and G represents circum centre, orthocentre and centroid respectively, then show
`HG:GO=2:1.` We have,

Answer

To show that \( HG : GO = 2 : 1 \), where \( O \), \( H \), and \( G \) represent the circumcenter, orthocenter, and centroid of a triangle respectively, we can follow these steps:

### Step 1: Understand the Definitions
- **Circumcenter (O)**: The point where the perpendicular bisectors of the sides of the triangle intersect.
- **Orthocenter (H)**: The point where the altitudes of the triangle intersect.
- **Centroid (G)**: The point where the medians of the triangle intersect. The centroid divides each median in the ratio \( 2:1 \).

### Step 2: Draw the Triangle
Let’s consider triangle \( ABC \) and plot points \( O \), \( H \), and \( G \) on it. Draw the triangle and label the vertices \( A \), \( B \), and \( C \).

### Step 3: Draw the Altitude and Median
- Draw the altitude from vertex \( A \) to side \( BC \) and label the foot of the altitude as point \( D \).
- Draw the median from vertex \( A \) to the midpoint \( E \) of side \( BC \).

### Step 4: Identify the Points
- Mark the orthocenter \( H \) at the intersection of the altitudes.
- Mark the centroid \( G \) at the intersection of the medians.
- Mark the circumcenter \( O \) at the intersection of the perpendicular bisectors.

### Step 5: Prove Triangles Congruence
To find the ratio \( HG : GO \), we will show that triangles \( AHG \) and \( EGO \) are similar:
- **Angle \( HAG \) is equal to angle \( EGO \)**: These angles are alternate interior angles.
- **Angle \( AGH \) is equal to angle \( GEO \)**: These angles are vertically opposite angles.

By the AA criterion, triangles \( AHG \) and \( EGO \) are similar.

### Step 6: Use the Properties of the Centroid
Since \( G \) is the centroid, it divides the median \( AE \) into two segments:
- \( AG = 2 \cdot GE \)

### Step 7: Set Up the Proportions
From the similarity of triangles \( AHG \) and \( EGO \):
\[ \frac{AG}{EG} = \frac{HG}{GO} \]
Substituting \( AG = 2 \cdot GE \):
\[ \frac{2 \cdot GE}{GE} = \frac{HG}{GO} \]
This simplifies to:
\[ 2 = \frac{HG}{GO} \]

### Step 8: Conclude the Ratio
Thus, we find that:
\[ HG : GO = 2 : 1 \]