Write Up 8: Altitudes and Orthocenters

Introduction:

In this exploration, we are going to investigate relationships between angles measures of triangles through the scope of triangles created by orthocenters and circumcenters.

Construction:

Step 1. Lets construct a general acute triangle ABC with its orthocenter labeled H:

The orthocenter was construct by constructing the angle bisectors of each interior angle of the triangle ABC. All of these angle bisectors meet at one intersection point called the orthocenter. This point is labeled H.

Step 2: Construct the circumcenter of triangle ABC and labeled this point C. Also construct the circle with radius AC. This circle with radius AC is the circumcircle of triangle ABC:

The circumcircle is constructed by finding the intersection point of all of the perpendicular bisectors of each side of triangle ABC. All of these perpendicular bisector intersect at one point called the circumcenter labeled C. Then by constructing a circle with radius AC, we are given the circumcircle centered at point C. (Green circle above)

Step 3: Now we can notice that the angle bisectors of the interior angles of triangle ABC intersect the circumcircle at 3 points. Lets label these points as L, M, and N. By connecting these points with line segments, we are given a new triangle LMN which is different from triangle ABC but shares the same circumcenter and circumcircle.

Angle Relationships Between ABC and LMN:

After exploring and investigating with the interior angles of triangle LMN and triangle ABC, I have found an interesting relationship amount the interior angles. I first started exploring sum on the angles between different the different triangles. I found that if we take two consecutive angles from triangle ABC, say angles A and B, and we take half the sum of these angles. The calculation that we get is equal to the interior angle measure of the part of triangle LMN than lies in between angles A and B. In the picture above, the interior angle N lies in between angles A and B. So mathematically, we get:

Lets see this relationship on GSP:

Conclusion:

The relationship between the interior angles of triangle ABC and triangle LMN stays consistent when manipulating triangle ABC. Therefore when triangle ABC is obtuse, the calculationis still true. The calculation si also true for when triangle ABC is a right triangle. Another conclusion that I found is that the relationship is not communative. Therefore, the calculation is not true. In conclusion, the relationship only hold when the sum of the two interior angles are angles from triangle ABC.