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Orthocenter Calculator

Enter the coordinates of the triangle's vertices, and the calculator will instantly determine the coordinates of its orthocenter.

Orthocenter A:

x₁

y₁

Orthocenter B:

x₂

y₂

Orthocenter C:

x₃

y₃

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The online Orthocenter Calculator allows you to find the orthocenter of any triangle quickly and accurately. This tool provides precise coordinates in seconds, helping students, teachers, and engineers visualize and understand triangle properties in trigonometry.

What is an Orthocenter?

The orthocenter is the point where the altitudes of a triangle intersect, also known as the point of concurrency.

Orthocenter Calculator

In the diagram above, the triangle has sides AB, BC, and CA. The corresponding altitudes are CF, AD, and BE. An altitude is a perpendicular line from a vertex to its opposite side. The intersection of all altitudes is the orthocenter, which can be calculated quickly using this calculator.

Properties of the Orthocenter

Acute Triangle: All angles are less than 90°. The orthocenter lies inside the triangle.

Acute Triangle

Obtuse Triangle: One angle is greater than 90°. The orthocenter lies outside the triangle.

Obtuse Triangle

Right Triangle: For a right triangle, the orthocenter coincides with the vertex of the right angle.

Orthocenter of Right Triangle

Algebraic Formulas to Find the Orthocenter

We use the slopes of sides and altitudes to calculate the orthocenter:

1. Slope of a side:

m = (y2 - y1) / (x2 - x1)

2. Slope of the perpendicular (altitude):

m_perpendicular = -1 / m

3. Equation of an altitude:

y - y1 = m_perpendicular × (x - x1)

Solving the equations of at least two altitudes simultaneously gives the orthocenter coordinates.

Example: Calculating the Orthocenter

Triangle vertices: A(2, -3), B(8, -2), C(8, 6)

Step 1: Slope of AC:

m_AC = (6 - (-3)) / (8 - 2) = 9 / 6 = 3/2

Step 2: Slope of altitude BE:

m_BE = -1 / (3/2) = -2/3

Step 3: Equation of altitude BE through B(8,-2):

y + 2 = (-2/3)(x - 8)

3(y + 2) = -2(x - 8)

3y + 6 = -2x + 16 → 2x + 3y - 10 = 0

Step 4: Slope of BC:

m_BC = (6 - (-2)) / (8 - 8) = 8 / 0 → undefined

Step 5: Slope of altitude AD through A(2,-3):

m_AD = 0 (perpendicular to vertical line)

Equation: y + 3 = 0 → y = -3

Step 6: Solve for x:

2x + 3(-3) - 10 = 0 → 2x - 19 = 0 → x = 9.5

Result: Orthocenter = (9.5, -3)

How the Orthocenter Calculator Works

Input:

Enter coordinates of all three vertices.
Click Calculate.

Output: The tool provides the exact orthocenter coordinates with a detailed step-by-step solution.

FAQs

Does the method change for different triangle types?

No. The process is the same for acute, obtuse, and right triangles.

Can a triangle exist without an orthocenter?

No. Every triangle has an orthocenter, since altitudes always intersect.

What does the orthocenter signify?

It represents the point of concurrency where all altitudes of the triangle meet.

What is Euler’s Line?

Euler’s Line passes through key triangle centers: the orthocenter, centroid, and circumcenter.

Conclusion

The orthocenter is a fundamental point of intersection in any triangle. Its position—inside, outside, or at a vertex—helps identify the triangle type. Using an orthocenter calculator ensures precise results instantly, making it a valuable tool for learning, teaching, and engineering applications.

References

Wikipedia: Orthocenter

Khan Academy: Common orthocenter and centroid

Lumen Learning: Rectangular Coordinate Systems and Graphs

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