A Triangle’s Orthocenter Given Its Coordinates

#

Adrian Jose Hernandez Vega

Introduction

#

The orthocenter of a triangle is defined as the point of intersection of all of the triangle’s altitudes.

An altitude of a triangle is a perpendicular segment connecting a vertex to the opposite side.

Suppose triangle \(\triangle ABC\)’s vertices have arbitrary coordinates \(A(x_1,y_1)\), \(B(x_2,y_2)\), and \(C(x_3,y_3)\). This triangle lies on the Cartesian coordinate plane.

Orthocenter Algebra

#

The altitudes of segments \(\overline{AB}\), \(\overline{BC}\), and \(\overline{CA}\) can be represented in slope-intercept form. If any two vertices form a perfectly vertical line, exclude such a segment, as it will lead to undefined results.

Lemma

#

Given slope \(m\) and point \((P_1,P_2)\), the line perpendicular to \(m\) that passes through \((P_1,P_2)\) is represented by:

\[ y = \frac{P_1 - x}{m} + P_2 \]

Proof: The outlined equation is a rephrasing of the point-slope formula, with the slope term inverted and negated to ensure perpendicularity.

The altitude perpendicular to \(\overline{AB}\) passes through \((x_3,y_3)\), the altitude perpendicular to \(\overline{BC}\) passes through \((x_1,y_1)\), and the altitude perpendicular to \(\overline{CA}\) passes through \((x_2,y_2)\). Using the previous lemma, these altitudes can be represented as:

\[ \begin{aligned} \overline{AB}_\perp \cap (x_3,y_3) &\implies y = \frac{(x_3-x)(x_2-x_1)}{y_2 - y_1} + y_3 \\ \overline{BC}_\perp \cap (x_1,y_1) &\implies y = \frac{(x_1-x)(x_3-x_2)}{y_3 - y_2} + y_1 \\ \overline{CA}_\perp \cap (x_2,y_2) &\implies y = \frac{(x_2-x)(x_3-x_1)}{y_3 - y_1} + y_2 \end{aligned} \]

Theorem

#

For triangle \(\triangle ABC\) with coordinates \(A(x_1,y_1)\), \(B(x_2,y_2)\), and \(C(x_3,y_3)\), the orthocenter can be found by solving any two non-vertical equations in the overdetermined system:

\[ \left\{ \begin{aligned} y &= \frac{(x_3-x)(x_2-x_1)}{y_2 - y_1} + y_3 \\ y &= \frac{(x_1-x)(x_3-x_2)}{y_3 - y_2} + y_1 \\ y &= \frac{(x_2-x)(x_3-x_1)}{y_3 - y_1} + y_2 \end{aligned} \right. \]