If the vertices P,Q,R of a triangle PQR are rational points, which of the following points of the triangle PQR is (are) always rational points(s)?

Options

A. centroid
B. incentre
C. circumcentre
D. orthocentre

Answer

To determine which points of triangle PQR are always rational points when the vertices P, Q, and R are rational points, we can analyze the properties of certain triangle centers: the centroid, orthocenter, and circumcenter.

### Step-by-Step Solution:

1. **Understanding Rational Points**:
- A point is considered a rational point if both its x-coordinate and y-coordinate are rational numbers. Given that the vertices P, Q, and R are rational points, we denote their coordinates as:
- P(x1, y1)
- Q(x2, y2)
- R(x3, y3)

2. **Finding the Centroid**:
- The centroid (G) of triangle PQR is calculated using the formula:
\[ G\left(\frac{x1 + x2 + x3}{3}, \frac{y1 + y2 + y3}{3}\right) \]
- Since the sum of rational numbers is rational, both coordinates of G will be rational. Thus, the centroid is always a rational point.

**Hint**: Check if the average of rational numbers remains rational.

3. **Finding the Orthocenter**:
- The orthocenter (H) is the point where the altitudes of the triangle intersect. The coordinates of the orthocenter can be derived from the equations of the altitudes, which depend on the slopes of the sides of the triangle.
- Since the slopes of the sides (formed by rational points) are rational, the equations of the altitudes will also have rational coefficients. Therefore, the orthocenter will also have rational coordinates.

**Hint**: Consider how the intersection of lines with rational coefficients behaves.

4. **Finding the Circumcenter**:
- The circumcenter (O) is the point where the perpendicular bisectors of the sides of the triangle intersect. The coordinates of the circumcenter can be calculated using the midpoints of the sides and the slopes of the sides.
- Since the midpoints and slopes are derived from the rational coordinates of the vertices, the circumcenter will also have rational coordinates.

**Hint**: Use the properties of perpendicular bisectors and midpoints to establish rationality.

5. **Conclusion**:
- Therefore, all three points (centroid, orthocenter, and circumcenter) of triangle PQR are always rational points when the vertices are rational points.

### Final Answer:
The points that are always rational points in triangle PQR are:
- Centroid
- Orthocenter
- Circumcenter