Orthocenter Calculator
Calculate orthocenter instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
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The orthocenter is found by computing two altitudes (lines from a vertex perpendicular to the opposite side) and finding their intersection. Each altitude slope is the negative reciprocal of the slope of the opposite side.
Last reviewed: December 2025
Worked Examples
Example 1: Orthocenter of an Acute Triangle
Example 2: Orthocenter of an Obtuse Triangle
Background & Theory
The Orthocenter Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Orthocenter Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Sources & References
Formula
Orthocenter = Intersection of altitudes (perpendicular from vertex to opposite side)
The orthocenter is found by computing two altitudes (lines from a vertex perpendicular to the opposite side) and finding their intersection. Each altitude slope is the negative reciprocal of the slope of the opposite side.
Worked Examples
Example 1: Orthocenter of an Acute Triangle
Problem: Find the orthocenter of the triangle with vertices A(0, 6), B(-3, 0), C(3, 0).
Solution: Slope of BC = (0 - 0)/(3 - (-3)) = 0 (horizontal line)\nAltitude from A perpendicular to BC is vertical: x = 0\nSlope of AC = (0 - 6)/(3 - 0) = -2\nAltitude from B perpendicular to AC has slope = 1/2\nEquation: y - 0 = 0.5(x - (-3)) => y = 0.5x + 1.5\nAt x = 0: y = 1.5\nOrthocenter = (0, 1.5)
Result: Orthocenter: (0, 1.5) — Inside the triangle (acute triangle)
Example 2: Orthocenter of an Obtuse Triangle
Problem: Find the orthocenter of the triangle with vertices A(0, 0), B(5, 0), C(1, 1).
Solution: Slope of BC = (1 - 0)/(1 - 5) = -0.25\nAltitude from A: slope = 4, equation: y = 4x\nSlope of AC = (1 - 0)/(1 - 0) = 1\nAltitude from B: slope = -1, equation: y = -(x - 5) = -x + 5\nIntersection: 4x = -x + 5 => 5x = 5 => x = 1, y = 4\nOrthocenter = (1, 4)
Result: Orthocenter: (1, 4) — Outside the triangle (obtuse triangle)
Frequently Asked Questions
What is the orthocenter of a triangle?
The orthocenter of a triangle is the single point where all three altitudes of the triangle intersect. An altitude is a line segment drawn from any vertex of the triangle perpendicular to the opposite side, or to the extension of that opposite side. Every non-degenerate triangle has exactly one orthocenter. The orthocenter is one of the four classical triangle centers alongside the centroid, circumcenter, and incenter. Unlike the centroid which always lies inside the triangle, the orthocenter can be inside, outside, or on the triangle depending on whether the triangle is acute, obtuse, or right respectively.
How do you calculate the orthocenter of a triangle?
To find the orthocenter, you need to determine the intersection point of at least two altitudes of the triangle. First, calculate the slope of one side of the triangle. The altitude from the opposite vertex has a slope that is the negative reciprocal of the side slope. Write the equation of this altitude line using the point-slope form with the vertex coordinates. Repeat the process for a second altitude using a different side and its opposite vertex. Then solve the two altitude line equations simultaneously to find their intersection point. That intersection is the orthocenter. The third altitude will automatically pass through the same point.
Where is the orthocenter located for different triangle types?
The location of the orthocenter depends entirely on the type of triangle. For an acute triangle, where all three interior angles are less than 90 degrees, the orthocenter lies strictly inside the triangle. For a right triangle, the orthocenter is located exactly at the vertex where the right angle is formed, because the two legs of the right triangle are themselves altitudes. For an obtuse triangle, where one angle exceeds 90 degrees, the orthocenter lies outside the triangle on the side opposite to the obtuse angle. This property makes the orthocenter unique among triangle centers.
What is the Euler line and how does the orthocenter relate to it?
The Euler line is a remarkable straight line that passes through several important triangle centers including the orthocenter, the centroid, and the circumcenter. The centroid always divides the segment from the orthocenter to the circumcenter in a ratio of two to one, with the centroid being closer to the circumcenter. This relationship was discovered by the Swiss mathematician Leonhard Euler in the 18th century. The Euler line demonstrates a deep geometric relationship between these triangle centers. In equilateral triangles, all three points coincide, so technically no unique Euler line exists for equilateral triangles.
What is the difference between orthocenter, centroid, circumcenter, and incenter?
These four points are the classical triangle centers but they have different definitions and properties. The centroid is the intersection of the three medians, which connect each vertex to the midpoint of the opposite side, and it always lies inside the triangle. The circumcenter is the intersection of the perpendicular bisectors of the three sides and is equidistant from all three vertices. The incenter is where the three angle bisectors meet and is equidistant from all three sides. The orthocenter is where the three altitudes intersect. Only the centroid and incenter are guaranteed to lie inside the triangle for all triangle shapes.
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References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy