How to find orthocenter coordinates?

Steps to Solve

1. Identify the Vertex Coordinates

Let the vertices of the triangle be labeled as $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$.

2. Determine the Slopes of the Sides

Calculate the slopes of the sides of the triangle. The slope of side $AB$ is given by:

$$ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} $$

The slope of side $AC$ is:

$$ m_{AC} = \frac{y_3 - y_1}{x_3 - x_1} $$

And the slope of side $BC$ is:

$$ m_{BC} = \frac{y_3 - y_2}{x_3 - x_2} $$

3. Find the Slopes of the Altitudes

The slopes of the altitudes can be found by taking the negative reciprocal of the slopes of the opposite sides. For the altitude from vertex $A$ to side $BC$, the slope is:

$$ m_{h_A} = -\frac{1}{m_{BC}} $$

For the altitude from vertex $B$ to side $AC$:

$$ m_{h_B} = -\frac{1}{m_{AC}} $$

For the altitude from vertex $C$ to side $AB$:

$$ m_{h_C} = -\frac{1}{m_{AB}} $$

4. Find the Equation of the Altitudes

Using the point-slope form of the line, write the equations for each of the altitudes. For example, the equation of the altitude from vertex $A$ can be expressed as:

$$ y - y_1 = m_{h_A}(x - x_1) $$

Do this similarly for the other two altitudes using their respective coordinates and slopes.

5. Solve the System of Equations

Solve the equations obtained from the altitudes. Choose any two altitude equations and set them equal to find their intersection point.

For example, solve the equations of the altitudes from $A$ and $B$ together:

$$ \begin{align*} y - y_1 &= m_{h_A}(x - x_1) \ y - y_2 &= m_{h_B}(x - x_2) \end{align*} $$

6. Calculate the Coordinates of the Orthocenter

Once you have the intersection point from the above step, you will have the coordinates of the orthocenter, which we'll denote as $H(h_x, h_y)$.