How do you find the orthocentre of a triangle and what is the formula used in this process?

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The orthocentre of a triangle is the point where all three altitudes intersect. To find the orthocentre, you can follow these steps:

  1. Identify the vertices: Start by labeling the vertices of the triangle. Let’s say the vertices are A, B, and C.
  2. Calculate the slopes: Determine the slopes of the sides of the triangle. For sides AB, BC, and CA, calculate the slopes as follows:
    • The slope of AB is (yB – yA) / (xB – xA)
    • The slope of BC is (yC – yB) / (xC – xB)
    • The slope of CA is (yA – yC) / (xA – xC)
  3. Find the slopes of the altitudes: The altitudes are perpendicular to the sides of the triangle, therefore their slopes are the negative reciprocal of the slopes of the respective sides:
    • The slope of the altitude from vertex C to side AB is -1/[(yB – yA) / (xB – xA)]
    • The slope of the altitude from vertex A to side BC is -1/[(yC – yB) / (xC – xB)]
    • The slope of the altitude from vertex B to side CA is -1/[(yA – yC) / (xA – xC)]
  4. Use point-slope form to find the equations of the altitudes: Using the slopes found in the previous step, write the equation of each altitude. For example, the altitude from C can be expressed as:
    • y – yC = mC(x – xC), where mC is the slope of the altitude from C.
  5. Solve the equations: To find the orthocentre, you need to solve the system of equations formed by any two altitude equations. The solution will give you the coordinates of the orthocentre (H).

Formula: While there is no simple one-formula method for finding the orthocentre directly, the orthocentre can also be calculated using the coordinates of the triangle’s vertices (A, B, C) with the formula:

H(x, y) = O (where O is a derived point based on circumradius and vertices.)

Finding the orthocentre requires some algebraic manipulation and visualizing the geometric relationships between the triangle’s altitudes.

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