Orthocenter Calculator

What Exactly IS an Orthocenter?

Okay, so imagine drawing a line from each corner of your triangle straight down (perpendicular) to the opposite side - these are called altitudes. Now here's the cool part: all three of these lines meet at ONE point! That point is the orthocenter. It's like they have a predetermined meeting spot. You only need to draw two altitudes to find it, but the third one will magically pass through the same point. Math is beautiful like that!

H = \textAltitude_A \cap \textAltitude_B \cap \textAltitude_C

How to Calculate It (With Coordinates)

If you know the three corner points of your triangle, you can find the orthocenter with some algebra. Here's the game plan: figure out the slope of two sides, flip and negative them to get perpendicular slopes (remember: perpendicular slopes multiply to give -1!), write equations for two altitudes, then solve where they intersect. That intersection? That's your orthocenter!

H(x_h, y_h) \text where m_h · m_side = -1

Where Does It Live? (Depends on Your Triangle!)

This is where it gets fun - the orthocenter's location tells you what kind of triangle you have:

• **Acute Triangle** (all angles < 90°): Orthocenter is INSIDE the triangle • **Right Triangle** (one 90° angle): Orthocenter sits right ON the 90° corner! • **Obtuse Triangle** (one angle > 90°): Orthocenter wanders OUTSIDE the triangle • **Equilateral Triangle** (all equal): Orthocenter is smack in the center with all the other special points

So if you find your orthocenter floating outside somewhere, you've got an obtuse triangle!

H_\textinside \Leftrightarrow \textacute, \quad H_\texton vertex \Leftrightarrow \textright, \quad H_\textoutside \Leftrightarrow \textobtuse

The Euler Line (Geometry's Coolest Pattern)

Ready for something mind-blowing? The orthocenter, centroid, and circumcenter all line up perfectly on a straight line called the Euler Line! It's named after Leonhard Euler who discovered this amazing pattern. And it gets better - they don't just randomly line up, they divide the line in a perfect 2:1 ratio. The centroid is exactly 2/3 of the way from the orthocenter to the circumcenter. Math doesn't do coincidences - when points line up like this, there's deep geometric harmony at work!

\vecG = \frac\vecO + 2\vecH3, \quad \overlineHG : \overlineGO = 2 : 1

Trilinear and Barycentric Coordinates

The orthocenter has elegant representations in homogeneous coordinate systems. In **trilinear coordinates** (distances from sides), the orthocenter is H = (\cos B \cos C : \cos C \cos A : \cos A \cos B) or equivalently (\tan A : \tan B : \tan C). In **barycentric coordinates** (weighted by vertices), the orthocenter is H = (\tan A : \tan B : \tan C) or (\cos B \cos C : \cos C \cos A : \cos A \cos B). These coordinates reveal the deep relationship between the orthocenter and the triangle's angles.

H_\texttrilinear = (\tan A : \tan B : \tan C)

The Nine-Point Circle

The nine-point circle is a remarkable circle associated with any triangle, passing through nine significant points: (1) the three feet of the altitudes (where altitudes meet opposite sides), (2) the three midpoints of the sides, and (3) the three midpoints of the segments from vertices to the orthocenter. The center of this circle, called the nine-point center N, lies exactly at the midpoint of the segment from orthocenter H to circumcenter O. The nine-point circle has radius exactly half the circumradius: r_9 = (R)/(2).

N = (H + O)/(2), \quad r_\textnine-point = (R)/(2)

The Orthic Triangle

The orthic triangle is formed by connecting the three feet of the altitudes (points H_A, H_B, H_C where altitudes meet opposite sides). This triangle has fascinating properties: (1) The original triangle is the orthic triangle of its orthic triangle. (2) The orthocenter of the original triangle is the incenter of the orthic triangle. (3) The sides of the orthic triangle have lengths |H_AH_B| = c|\cos C|, |H_BH_C| = a|\cos A|, |H_CH_A| = b|\cos B|. (4) The area of the orthic triangle is K_\textorthic = K · |\cos A \cos B \cos C| where K is the area of the original triangle.

K_\textorthic = K · |\cos A \cos B \cos C|

Distance Formulas

Several important distance relationships involve the orthocenter. The distance from orthocenter H to vertex A is |HA| = 2R|\cos A| where R is the circumradius. The distance from H to the midpoint of side BC is R√(1 - 8\cos A \cos B \cos C). The power of the orthocenter with respect to the circumcircle is R^2 - |OH|^2 = R^2(1 - 8\cos A \cos B \cos C). The distance from H to the circumcenter O is given by |OH|^2 = R^2(1 - 8\cos A \cos B \cos C).

|HA| = 2R|\cos A|, \quad |OH|^2 = R^2 - 2R^2(4\cos A \cos B \cos C)

Perpendicular Relationships

The orthocenter creates numerous perpendicular relationships. Each altitude is perpendicular to its corresponding side: altitude from A is perpendicular to BC, from B to AC, from C to AB. Additionally, the line from any vertex through the orthocenter is perpendicular to the opposite side of the orthic triangle. The reflection of the orthocenter over any side of the triangle lies on the circumcircle. These perpendicularity properties make the orthocenter essential in solving many geometric problems.

AH \perp H_BH_C, \quad BH \perp H_CH_A, \quad CH \perp H_AH_B

Reflection Properties

The orthocenter has beautiful reflection properties: (1) The reflection of H over any side of the triangle lies on the circumcircle. (2) The reflection of H over the midpoint of any side gives the point diametrically opposite to the corresponding vertex on the circumcircle. (3) The reflections of H over the three sides form a triangle homothetic to the original triangle with ratio -1 centered at the circumcenter. These properties connect the orthocenter to circle geometry.

\textReflect(H, \textside BC) \in \textcircumcircle

Applications in Advanced Geometry

The orthocenter appears in many advanced geometric theorems: (1) In the **Simson Line theorem**, the orthocenter helps determine pedal points. (2) In **triangle centers theory**, the orthocenter is the isogonal conjugate of the circumcenter. (3) The orthocenter is used in **spiral similarities** and **homothety** transformations. (4) In **analytic geometry**, the orthocenter simplifies many coordinate proofs. (5) The orthocenter appears in **physics** when analyzing triangular structures and their stability points.

\textOrthocenter applications in physics, engineering, and computer graphics

Worked Example

Inputs

• points: 0,0;4,0;2,3

Result: (2, 1.3333)

Explanation

For triangle with vertices A(0,0), B(4,0), C(2,3): • Side AB lies on the x-axis, so the altitude from C is the vertical line x = 2 • Side BC has slope (3-0)/(2-4) = -1.5, so the altitude from A has slope 2/3: y = (2/3)x • Substituting x = 2 gives y = 4/3 • Check with the third altitude from B (slope -2/3 through (4,0)): y = -2/3(2-4) = 4/3 ✓ • Orthocenter = (2, 1.3333)

Second Scenario

Inputs

• points: 0,0;4,0;2,3

Result: (2, 1.3333)

Explanation

This scenario uses different inputs (points = 0,0;4,0;2,3) to show how changing one variable affects the orthocenter result. Run the calculator above with these values to get the exact updated output with step-by-step work.