The vertices of a triangle are
`[at_(1)t_(2),a(t_(1)+t_(2))]`,`[at_(2)t_(3),a(t_(2)+t_(3))]`, `[at_(3)t_(1),a(t_(3)+t_(1))]`.
Find the orthocentre of the triangle.

Answer

To find the orthocenter of the triangle with vertices given by the coordinates \((at_1t_2, a(t_1 + t_2))\), \((at_2t_3, a(t_2 + t_3))\), and \((at_3t_1, a(t_3 + t_1))\), we will follow these steps:

### Step 1: Identify the vertices
Let:
- Vertex A: \(A(at_1t_2, a(t_1 + t_2))\)
- Vertex B: \(B(at_2t_3, a(t_2 + t_3))\)
- Vertex C: \(C(at_3t_1, a(t_3 + t_1))\)

### Step 2: Find the slope of line BC
The slope \(m_{BC}\) of line segment BC can be calculated using the formula:
\[ m_{BC} = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of points B and C:
\[ m_{BC} = \frac{a(t_2 + t_3) - a(t_3 + t_1)}{at_2t_3 - at_3t_1} \]
This simplifies to:
\[ m_{BC} = \frac{a(t_2 - t_1)}{a(t_2t_3 - t_3t_1)} = \frac{t_2 - t_1}{t_2t_3 - t_3t_1} \]

### Step 3: Find the slope of line AC
Similarly, the slope \(m_{AC}\) of line segment AC is:
\[ m_{AC} = \frac{a(t_1 + t_2) - a(t_3 + t_1)}{at_1t_2 - at_3t_1} \]
This simplifies to:
\[ m_{AC} = \frac{a(t_2 - t_3)}{a(t_1t_2 - t_3t_1)} = \frac{t_2 - t_3}{t_1t_2 - t_3t_1} \]

### Step 4: Find the slope of the altitude from A to BC
The slope of the altitude from A to line BC, which is perpendicular to BC, is given by:
\[ m_{altitudeA} = -\frac{1}{m_{BC}} = -\frac{t_2t_3 - t_3t_1}{t_2 - t_1} \]

### Step 5: Write the equation of the altitude from A
Using point-slope form, the equation of the altitude from A is:
\[ y - a(t_1 + t_2) = m_{altitudeA}(x - at_1t_2) \]

### Step 6: Find the slope of the altitude from B to AC
Similarly, the slope of the altitude from B to line AC is:
\[ m_{altitudeB} = -\frac{1}{m_{AC}} = -\frac{t_1t_2 - t_3t_1}{t_2 - t_3} \]

### Step 7: Write the equation of the altitude from B
The equation of the altitude from B is:
\[ y - a(t_2 + t_3) = m_{altitudeB}(x - at_2t_3) \]

### Step 8: Solve the equations of the altitudes
To find the orthocenter, we need to solve the two equations obtained from the altitudes. By substituting and eliminating \(y\), we can find the \(x\) coordinate of the orthocenter.

### Step 9: Substitute back to find \(y\)
Once we have \(x\), substitute it back into one of the altitude equations to find the corresponding \(y\) coordinate.

### Final Result
After performing the calculations, we find that the orthocenter \(H\) of the triangle has coordinates:
\[ H\left(-a, a(t_1 + t_2 + t_3 + t_1t_2t_3)\right) \]