Recall that 3 or more lines are said to be concurrent if and only if they intersect at exactly 1 point.
The angle bisectors of a triangle's 3 interior angles are all concurrent.
Their point of concurrency is called the INCENTER of the triangle.
In the applet below, point I is the triangle's INCENTER. Use the tools of GeoGebra in the applet below to complete the activity below the applet. Be sure to answer each question below the applet fully as you proceed.
The angle bisectors of a triangle's 3 interior angles are all concurrent.
Their point of concurrency is called the INCENTER of the triangle.
In the applet below, point I is the triangle's INCENTER. Use the tools of GeoGebra in the applet below to complete the activity below the applet. Be sure to answer each question below the applet fully as you proceed.
1) Click the checkbox that says "Drop Perpendicular Segments from I to sides.
2) Now, use the Distance tool to measure and display the lengths IG, IH, and IJ. What do you notice?
3) Experiment a bit by moving any one (or more) of the triangle's vertices around Does your initial observation in (2) still hold true? Why is this?
4) Construct a circle centered at I that passes through G. What else do you notice Experiment by moving any one (or more) of the triangle's vertices around. This circle is said to be the triangle's incircle, or inscribed circle. It is the largest possible circle one can draw inside this triangle. Why, according to your results from (2), is this possible?
5) Do the angle bisectors of a triangle's interior angles also bisect the sides opposite theses angles? Use the Distance tool to help you answer this question.
6) Is it ever possible for a triangle's INCENTER to lie OUTSIDE the triangle If so, under what condition(s) will this occur?
7) Is it ever possible for a triangle's INCENTER to lie ON the triangle itself? If so, under what condition(s) will this occur?
2) Now, use the Distance tool to measure and display the lengths IG, IH, and IJ. What do you notice?
3) Experiment a bit by moving any one (or more) of the triangle's vertices around Does your initial observation in (2) still hold true? Why is this?
4) Construct a circle centered at I that passes through G. What else do you notice Experiment by moving any one (or more) of the triangle's vertices around. This circle is said to be the triangle's incircle, or inscribed circle. It is the largest possible circle one can draw inside this triangle. Why, according to your results from (2), is this possible?
5) Do the angle bisectors of a triangle's interior angles also bisect the sides opposite theses angles? Use the Distance tool to help you answer this question.
6) Is it ever possible for a triangle's INCENTER to lie OUTSIDE the triangle If so, under what condition(s) will this occur?
7) Is it ever possible for a triangle's INCENTER to lie ON the triangle itself? If so, under what condition(s) will this occur?
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.
Drag the white vertices of the triangle around and then use your observations to answer the questions that appear below the applet.
Consider the following questions based on the applet above:
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.)
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.)