Using GeoGebra, draw an acute-angled, non-equilateral \(\Delta ABC\) and construct the circumcenter, \(K\). Once you have constructed your circumcenter, hide your lines.
Measure \(\angle A\) and \(\angle BKC\), \(\angle B\) and \(\angle CKA\), and \(\angle C\) and \(\angle AKB\). Move all the angle measurements so that you can read them. What do you conjecture is the relationship between these angles?
Select side \(AB\) and construct its median. Then, click on side \(AB\) and see what its name is. In the input bar, create a new variable, \(L\), that is half the length of \(AB\). Type L = and then the name of side \(AB\) divided by two.
Click on the median you created to figure out its name. Then, create a text box that shows the length of the median and the length of your new variable \(L\). Move the vertices of \(\Delta ABC\) so that the median is equal to \(L\). What is special about this triangle? What can be said about the median to the hypotenuse of a right triangle?
Is it true that the midpoint of the hypotenuse of a right triangle is equidistant to all three vertices? If so, why? If not, give a counterexample.
