Exploring the Circumcircle
- Warren Koepp
Interactive sketch based on the exploration of the circumcircle of triangle ABC that we did in class to introduce Chapter 2. The original triangle is in black, with sides extended as dotted lines. Interior angle bisectors are solid, exterior angle bisectors are dashed, with color codes: red for vertex A, blue for B, and green for C. The perpendicular bisectors of the sides are in purple, with A' and A'' (respectively) marking the intersection of the interior and exterior angle bisectors at A with the perpendicular bisector of the opposite side (BC). Points B', B'', C', and C'' are similarly defined. Three points of concurrence are marked consistent with the usage in our text: the circumcenter O (perpendicular bisectors of sides), the orthocenter H (altitudes, in yellow), and the incenter I (interior angle bisectors). The centroid G (medians) does not appear here. Three additional points of concurrence of 3 lines are marked. These are L, M, and K, the excenters--points at which the exterior angle bisectors from 2 vertices meet the interior angle bisector from the third vertex. The points H', H'', and H''' are the reflections of the orthocenter (point of concurrence of the three altitudes) across each of the three sides of the original triangle ABC.
There are lots of possible theorems that jump out in this picture, regarding points of concurrence of 3 lines, and collections of more than three points lying on a common circle. The circle in the picture is determined by A, B, and C. How many of the other points can you prove must lie on this circle? Which of the sets of three lines have we proved to be concurrent at a single point? Can you prove the others? As you manipulate the picture you may add features or hide features. Can you frame any particular conjectures that we might try to prove? Be as precise as you can in your statements--it is important to know exactly what you are trying to prove!