This exploration will examine the ORTHOCENTER (H) of various triangles. The ORTHOCENTER or orthogonal center is where all of the altitudes of the triangles intersect. The respective altitudes pass through a given vertex and intersect the side opposite that vertex at a right angle. An example of an altitude is given below:

Note: The symbol ( ) shall be used to mean congruent.

Please note that in the following cases it is not a strict requirement that H lie on the inside of the triangle, but it may lie on the extensions of the altitudes outside the triangle.

Let us first start with an equilateral triangle (Case I: all sides congruent AB AC BC):

In this case H lies in the center of triangle ABC and does not lie anywhere else.

Case II (isosceles triangle- two sides congruent AB AC)

The altitudes are constructed by creating the perpendicular lines from a side to the vertex opposite that side. For example altitude A is constructed by creating a line perpendicular to segment BC and passing through A. This construction may be slightly different for some triangles as we may see…

This seems to be similar to the case for the equilateral triangle except H is closer to the side which is not congruent to the other two.

Case III (scalene triangle - all sides of different lengths):

Subcase A: (obtuse triangle-one angle measure is more than 90^o)

In this case, the orthocenter M lies outside the triangle with the following correspondences:

Altitude AM is orthogonal to seg (CB),

Altitude BM is orthogonal to seg (AC),

Altitude CM is orthogonal to seg (AB).

 

Please note that in all cases the line segment altitude is drawn, instead of the entire line.

Subcase B: (Acute triangle – no angle measure greater than 90^o)

In this case we can again see that the orthocenter is inside of the triangle. Please note that my use of notation M or H is arbitrary.

Interesting Case (right triangle): In a right triangle two of the sides are orthogonal to one another, that is, they meet at 90^o. Now when thinking about how we constructed the orthocenter, we created perpendicular line to a vertex and the vertices’ respective opposite side.

Altitude A is perpendicular to CM so, ABĂ line AB, similarly for altitude C being perpendicular to AB so, CMĂ line CM. The only place where these two lines intersect is at point B. When constructing the third altitude B (perpendicular to AC), we see that it obviously passes through B! In this interesting case the orthocenter is one of the vertices, specifically, the vertex of the occurrence of the right triangle.

What can you come up with?