The orthocentre of a triangle is the point where all three altitudes intersect. To find the orthocentre, follow these steps:
- Identify Vertices: Label the vertices of the triangle as A, B, and C.
- Find the Slopes: Determine the slopes of the sides of the triangle. For sides AB, BC, and CA, calculate the slopes using the coordinates of the points:
- Slope of AB = (y2 – y1) / (x2 – x1)
- Slope of BC = (y3 – y2) / (x3 – x2)
- Slope of CA = (y1 – y3) / (x1 – x3)
- Calculate the Perpendicular Slopes: The slopes of the altitudes will be the negative reciprocals of the slopes of the sides:
- Altitude from A to BC slope = -1/(slope of BC)
- Altitude from B to CA slope = -1/(slope of CA)
- Altitude from C to AB slope = -1/(slope of AB)
- Write Equations of the Altitudes: Using point-slope form, write the equations of the altitudes:
- Altitude from A: y – y1 = (slope of altitude)(x – x1)
- Altitude from B: y – y2 = (slope of altitude)(x – x2)
- Find Intersections: Solve the equations of any two altitudes simultaneously to find the point of intersection, which is the orthocentre H.
Formula: Although there is no simple formula to find the orthocentre directly from coordinates like there is for the centroid or circumcentre, the intersection of the altitudes determines it. However, if coordinates are known, the coordinates of the orthocentre H can be calculated using the formula:
H = (x1 + x2 + x3, y1 + y2 + y3) – 2 * ( (A + B + C) / 3 )
where A, B, and C are the areas of the triangles formed with the orthocentre and the vertices of the triangle.