Circumcenter (& Questions)
Recall that 3 or more lines are said to be concurrent if and only if they intersect at exactly 1 point.
A triangle's 3 perpendicular bisectors are concurrent. Their point of concurrency is called the CIRCUMCENTER of the triangle.
In the applet below, point C is the circumcenter of the triangle. Move the white vertices of the triangle around and then use your observations to answer the questions that appear below the applet.
Questions:
1) Is it ever possible for a triangle's circumcenter to lie OUTSIDE the triangle? If so, under what circumstance(s) will this occur?
2) Is it ever possible for a triangle's circumcenter to lie ON THE TRIANGLE ITSELF? If so, under what circumstance(s) will this occur?
3) If your answer for (2) was "YES", where on the triangle did point C lie?
4) Is it ever possible for a triangle's circumcenter to lie INSIDE the triangle? If so, under what circumstance(s) will this occur?
5) Now, on the applet above, construct a circle centered at C that passes through J. What do you notice? (Hint: Look at points K & L.)
6) Let's generalize: The circumcenter of a triangle is the ONLY POINT that is.............(If you need a hint to complete this step, consider the lengths CK & CL with respect to length CJ.)
7) Take a look back the worksheet found here.
Use your observations from examining this worksheet again to explain why the phenomenon you
observed in step (5) and your response to this phenomenon (in step 6) occurred.