The purpose of this write up is to explore relationships of orthocenters constructed from creating three triangles from the using the orthocenter of the original triangle as a vertex. I will also examine the circumcircles and circumcenters of these triangles. The first step is to construct triangle ABD and it's orthocenter, H. Recall that the orthocenter is the intersection of of the altitudes of the triangle.
Next, we will construct the orthocenters of triangles HBD, HAD, and HAB.
Notice, that the intersection of the altitudes of triangle HBD is vertex A of the original triangle. This means that the orthocenter of triangle HBD is vertex A. Will this happen for all three triangles?
This phenomenon also occurs for triangle HAD. The orthocenter of triangle HAD is vertex B or the original triangle.
Again, the orthocenter of triangle HAB is vertex D of the original triangle. My conjecture is that this occurs for all triangles constructed in this manner. To manipulate a triangle constructed with 2 vertices of the original triangle and the orthocenter of the original triangle as the third vertex, click here. Choose a point, then drag the point around, changing the size of the triangle. Notice that no matter how the triangle changes, the vertex of the original triangle which was not a vertex of the new triangle continues to be the orthocenter of the new triangle. This is a good indication that my conjecture is correct. Note, you must have Geometers Sketch Pad in order to manipulate the triangle.
My next goal is to construct the circumcircles of triangles ABD, HBD, HAB, and HAD. Recall that the circumcircle of a triangle is the triangle with the circumcenter of the triangle as the center of the circle and all three vertices of the triangle lie on the circle. The circumcenter of a triangle is the intersection of the perpendicular bisectors of the triangle.
The circumcenter of triangle ABD is labeled C1, and the circumcenters of triangles HAD, HBD, and HAB are C2, C3, and C4 respectively. Are there any relationships between the circumcenters of each triangle?
Using GSP to measure the distance of each circumcenters C2 - C4 to their two closest vertices of the original triangle ABD, we get
Clearly, the circumcenters of each triangle, constructed with the orthocenter, H, as a vertex, are equidistant from the two closest vertices of the original triangle ABD. This forms a hexagon with congruent sides, but the hexagon is not regular since it is not equal-angular. We can also see, by using GSP to measure the distance, that the circumcenter of triangle ABD is the same distance from all three vertices.
What would happen if any vertex of triangle ABD was moved to where the orthocenter H is located? In the picture below, the vertex B has been moved to H. Has anything special occured?
Using GSP to measure the angle ABD, we get 90 degrees. Does this always happen? Can we prove that if the orthocenter is a vertex of the triangle, then the triangle must be a right triangle?
The proof follows by the definition of orthocenter. Remember that the orthocenter is the intersection of the altitudes of the triangle. Since the altitude from vertex A to side BD is perpendicular to side BD, by definition of altitude, and B has moved to the same location as H, then B lies on the altitude from A to side BD. This means that the angle ABD must be a right angle. Therefore triangle ABD must be a right triangle if B is the orthocenter of ABD.