1. What is an Orthocentre?

The orthocentre is the intersection point of the three altitudes of a triangle. An altitude is a perpendicular line from a vertex to the opposite side (or its extension).

2. How Does the Calculator Work?

The calculator finds the orthocentre by:

Mathematical Process:

• Slope of AB: \( m_{AB} = \frac{y2 - y1}{x2 - x1} \)
• Slope of altitude from C: \( m_{C} = -\frac{1}{m_{AB}} \)
• Equation of altitude from C: \( y - y3 = m_C(x - x3) \)
• Similarly for other altitudes
• Solve two altitude equations to find intersection

3. Importance of Orthocentre

Details: The orthocentre is one of the triangle's notable points, along with centroid, circumcentre, and incentre. It has applications in geometry, engineering, and computer graphics.

4. Using the Calculator

Tips: Enter coordinates of three non-collinear points. The calculator will find the orthocentre coordinates. For best results, avoid points that form a right-angled triangle (where orthocentre coincides with a vertex).

5. Frequently Asked Questions (FAQ)

Q1: Where is the orthocentre located in different triangles?
A: In acute triangles: inside the triangle. In right triangles: at the right-angled vertex. In obtuse triangles: outside the triangle.

Q2: Can three collinear points have an orthocentre?
A: No, collinear points don't form a triangle and thus have no orthocentre.

Q3: How is orthocentre related to other triangle centers?
A: In an equilateral triangle, orthocentre coincides with centroid and circumcentre. In any triangle, orthocentre, centroid and circumcentre are colinear (Euler line).

Q4: What if two altitudes are parallel?
A: This happens in right-angled triangles where two altitudes are the legs, and the orthocentre is at the right angle vertex.

Q5: Can the orthocentre be at infinity?
A: No, though in degenerate cases (like three colinear points) the concept doesn't apply.