Orthocenter Calculator

An orthocenter is the point of concurrency of a triangle's three altitudes—line segments drawn from each vertex perpendicular to the opposite side. Mathematically, the coordinates of the orthocenter (X_ortho, Y_ortho) can be found using the trilinear coordinates or by determining the equations of at least two altitude lines and solving for their intersection. Using the vertices A(x1, y1), B(x2, y2), and C(x3, y3), the calculator first finds the slope of the triangle's sides, such as the AB slope and BC slope. It then utilizes the negative reciprocal of these slopes to formulate the perpendicular altitude lines. Alternatively, using trigonometric barycentric weights involving the angles at each vertex, the coordinates are computed as: X_ortho = (x1 * tan(A) + x2 * tan(B) + x3 * tan(C)) / (tan(A) + tan(B) + tan(C)), and similarly for Y_ortho.

Consider a triangle with vertices defined at coordinate points A(0, 0), B(4, 0), and C(0, 3), forming a classic right-angled triangle. First, we find the lengths of the sides using the distance formula: side a equals 5, side b equals 4, and side c equals 3. Next, we determine the slopes of the sides. The slope of AB is 0, and the slope of BC is -3/4. The altitude from vertex C to side AB is a vertical line passing through x = 0. The altitude from vertex A to side BC must be perpendicular to BC (slope 4/3) and pass through (0,0), giving the line equation y = (4/3)x. Solving the system of equations where x = 0 and y = (4/3)x yields the intersection point at (0, 0). Thus, for this right triangle, the orthocenter is located directly at vertex A(0, 0).

When inputting coordinates, always double-check your sign values (positive and negative) to prevent skewed slope calculations. Remember that for obtuse triangles, the orthocenter will naturally fall outside the physical boundary of the triangle, which is a mathematically correct result rather than an error. If you are working with side slopes rather than raw coordinates, ensure your slope difference inputs are accurately calculated before running advanced transformations.