# Incenter Exploration (A)

- Author:
- Cheris South, Tim Brzezinski

- Topic:
- Angles, Circle, Incircle or Inscribed Circle

**NCENTER**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 fully as you proceed.*

**Directions:**

1) In the applet above construct a line passing through I and is perpendicular to *AB*.
2) Use the **Intersect** tool to plot and label a point *G* where the line you constructed in (1) intersects *AB*.
3) Construct a line that passes through I and is perpendicular to *BC*.
4) Plot and label a point *H* where the line you constructed in (3) intersects *BC*.
5) Construct a line that passes through I and is perpendicular to *A**C*.
6) Plot and label a point *J* where the line you constructed in (5) intersects *AC*.
7) Now, use the **Distance** tool to measure and display the lengths *IG*, *IH*, and *IJ*. What do you notice?
8) Experiment a bit by moving any one (or more) of the triangle's vertices around
Does your initial observation in (7) still hold true?
Why is this? (If you need a hint, refer back to the worksheet found here.

*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 (7) is this possible? 10) 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. 11) Is it ever possible for a triangle's

**INCENTER**to lie OUTSIDE the triangle? If so, under what condition(s) will this occur? 12) Is it ever possible for a triangle's

**INCENTER**to lie ON the triangle itself? If so, under what condition(s) will this occur?

**Directions:**

1) In the applet above construct a line passing through I and is perpendicular to *AB*.
2) Use the **Intersect** tool to plot and label a point *G* where the line you constructed in (1) intersects *AB*.
3) Construct a line that passes through I and is perpendicular to *BC*.
4) Plot and label a point *H* where the line you constructed in (3) intersects *BC*.
5) Construct a line that passes through I and is perpendicular to *A**C*.
6) Plot and label a point *J* where the line you constructed in (5) intersects *AC*.
7) Now, use the **Distance** tool to measure and display the lengths *IG*, *IH*, and *IJ*. What do you notice?
8) Experiment a bit by moving any one (or more) of the triangle's vertices around
Does your initial observation in (7) still hold true?
Why is this? (If you need a hint, refer back to the worksheet found here.

*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 (7) is this possible? 10) 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. 11) Is it ever possible for a triangle's

**INCENTER**to lie OUTSIDE the triangle? If so, under what condition(s) will this occur? 12) Is it ever possible for a triangle's

**INCENTER**to lie ON the triangle itself? If so, under what condition(s) will this occur?