Circumcenter (& Questions)
[color=#000000]Recall that 3 or more lines are said to be concurrent if and only if they intersect at exactly 1 point.[br][br]A triangle's 3 [/color][color=#cc0000][b]perpendicular bisectors[/b] [/color][color=#000000]are concurrent. Their point of concurrency is called the [/color][b][color=#cc0000]CIRCUMCENTER[/color][/b][color=#000000] of the triangle. [br][/color][br][color=#000000]In the applet below, [/color][color=#cc0000][b]point C[/b][/color][color=#000000] is the [/color][b][color=#cc0000]circumcenter[/color][/b][color=#000000] 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.[/color]
[color=#000000]Questions:[br][br]1) Is it ever possible for a triangle's [/color][color=#cc0000][b]circumcenter[/b][/color][color=#000000] to lie OUTSIDE the triangle? If so, under what circumstance(s) will this occur?[br][br]2) Is it ever possible for a triangle's[/color] [b][color=#cc0000]circumcenter[/color][/b] [color=#000000]to lie ON THE TRIANGLE ITSELF? If so, under what circumstance(s) will this occur?[br][br]3) If your answer for (2) was "YES", where on the triangle did [/color][color=#cc0000][b]point C[/b][/color][color=#000000] lie?[br][/color][br][color=#000000]4) Is it ever possible for a triangle's[/color] [color=#cc0000][b]circumcenter[/b][/color] [color=#000000]to lie INSIDE the triangle? If so, under what circumstance(s) will this occur?[br][br]5) Now, on the applet above, construct a circle centered at[/color][color=#cc0000][b] C[/b][/color] [color=#000000]that passes through J. What do you notice? ([i]Hint: Look at points K & L.[/i]) [br][br]6) Let's generalize: The[/color] [color=#cc0000][b]circumcenter[/b][/color] [color=#000000]of a triangle is the [/color][b][color=#cc0000]ONLY POINT[/color][/b][color=#000000] that is.............[/color][i][color=#000000](If you need a hint to complete this step, consider the lengths CK & CL with respect to length CJ.)[br][br][/color][/i][color=#000000]7) Take a look back the worksheet found [url=https://tube.geogebra.org/m/TW9C7Y4k][color=#000000]here[/color][/url]. [br] Use your observations from examining this worksheet again to explain why the phenomenon you [br] observed in step (5) and your response to this phenomenon (in step 6) occurred. [/color]