Orthocenter of a triangle

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The orthocenter is the intersection of the three heights of a triangle, which can be found inside or outside the figure.

It should be remembered that the height of a triangle is that segment that starts from each vertex of the triangle and extends towards its opposite side, forming a right angle or 90º. That is, the height and its respective side are perpendicular.

In the figure above, for example, point O is the orthocenter of the figure, with the heights of the triangle being CF, BE, and AD.

Orthocenter according to the type of triangle

The orthocenter, depending on the type of triangle in question, has different characteristics:

  • Right triangle: The orthocenter of a right triangle coincides with the vertex that corresponds to the right angle. In the figure below, for example, the heights are BF and the triangle segments AB and BC themselves, the orthocenter being the vertex B.

It is also worth mentioning that the heights AB and BC are the legs, that is, the sides that form the right angle, while AC is the hypotenuse.

  • Obtuse triangle: The orthocenter is outside the triangle when it is obtuse, that is, when one of the interior angles of the figure is greater than 90º.

In the image below, for example, the heights are AH, CI and FB, so we look for the point of intersection of their extensions, which would be point O.

  • Acute triangle: The orthocenter is located inside the figure when the triangle is acute, that is, when all its internal angles are acute or less than 90º (see the first image of this article).

Orthic triangle

The orthic triangle is one whose vertices are the feet of the three heights of the triangle. As we see in the figure below, the orthic triangle of triangle ABC is triangle FGH.

It is also true that the orthocenter (point I) of triangle ABC is also the center of the inscribed circle (contained in) the orthic triangle.

How to find the orthocenter of a triangle

Suppose we have the equation of the lines that contain two of the heights of a triangle that are the following:

y = -137.7x-1941

y = 0.6x + 7

So, we must find at what values ​​of x and y both lines coincide. First we solve for x by equating the right side of each equation:

-137.7x-1941 = 0.6x + 7

-138.3x = 1948

x = -14.0853

Then, we solve for and in either of the two equations:

y = (0.6x-14.0853) +7

y = -8.4512 + 7 = -1.4512

Therefore, the coordinates of the orthocenter in the Cartesian plane are (-14.0853, 1.4512)

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