Altitudes and Orthocenters

This investigation starts with any triangle ABC and examines the triangle formed by the points where the extended altitudes meet the circumcircle. Then this triangle is compared to the orthic triangle.

Let's begin by constructing any triangle ABC and its extended altitudes. Remember that an altitude is the perpendicular line drawn from a vertex to the opposite side of a triangle. The intersection of the three altitudes is called the orthocenter ; we label it H.

Let's connect the feet of the altitudes to construct triangle DEF. This is the orthic triangle.


Now we need the circumcircle. First we find the circumcenter of the triangle ABC. It is the point which is the same distance from each vertex. To find it, we construct the perpendicular bisectors of each side of the triangle; they are colored blue to distinguish them from the extended altitudes. The bisectors meet at Ci.


Notice that the perpendiculaar bisectors are all parallel to the altitudes. Why? Now Ci is the same distance from each vertex and is the center of the circumcircle. Let's use the distance from Ci to C to be the radius of the circumcircle.

Now we'll hide the perpendicular bisectors to make viewing a little easier. Notice that each extended altitude intersects the circumcircle in two points. One is at the vertex; recall that each vertex is the distance of a radius from the circumcenter. The other is the intersection we've been working for to construct the final triangle, RST.


Triangle RST certainly appears to be similar, that is it has the same shape and same size angles, as triangle DEF.

Click here to see a GSP sketch which can be manipulated to show the relationships of the triangle by moving A,B, or C along the circumcircle.

The end!