Step 1 Construct the angle bisector of angle A.[br][br]Step 2 Construct the bisector of angle B.[br][br]Step 3 Construct point D where these two rays intersect.[br][br]Investigate[br][br]1. Construct the bisector of angle C. What do you notice about this third angle bisector? Drag the vertices of the triangle to see if this is always true.[br][br]2. If you were blindfolded and drew three lines on a piece of paper, do you think they would intersect in a single point? When lines that lie in the same plane do intersect in a single point, they’re said to be concurrent. Complete the Angle Bisector Concurrency Conjecture (C-9) in your conjecture packet.[br][br]3. The point of concurrency of the angle bisectors in a triangle is called the incenter. Measure the distance from point D to AB. How did you do this? Also measure the distances from point D to the other two sides. Drag parts of the triangle; complete the Incenter Conjecture (C-13) in your conjecture packet.

Step 1 Construct points D and E, the midpoints of segment AB and segment AC.[br][br]Step 2 Construct the perpendicular bisector of segment AB.[br][br]Step 3 Construct the perpendicular bisector of segment AC.[br][br]Step 4 Construct point F where these two perpendicular bisectors intersect,[br]and then construct the perpendicular bisector of segment BC.[br][br]Investigate[br][br]1. Complete the Perpendicular Bisector Concurrency Conjecture (C-10) in your conjecture packet.[br][br]2. The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter. Measure distances from the circumcenter to each of the three vertices of the triangle. Drag parts of the triangle;[br]complete the Circumcenter Conjecture (C-12) in your conjecture packet.

Step 1 Construct the perpendicular line from point B to side AC. This line contains an altitude of triangle ABC.[br][br]Step 2 Construct a line through point A that contains an altitude.[br][br]Step 3 Construct point D, the point of intersection of these two lines, and then construct the line containing the third altitude in triangle ABC, through point C[br][br]Investigate[br][br]1. Complete the Altitude Concurrency Conjecture (C-11) in your conjecture packet.

Step 1 Construct the midpoint D of AC and median BD.[br][br]Step 2 Construct the midpoint E of AB and median CE.[br][br]Step 3 Construct point F where these medians intersect, and then construct the third median, from point A to side BC.[br][br]Investigate[br][br]1. Complete the Median Concurrency Conjecture (C-14) in your conjecture packet.

Step 1 Change the label of point F to Ce, for centroid. The centroid is the point of concurrency of the medians.[br][br]Step 2 Measure the distance from B to Ce and the distance from Ce to D.[br][br]Step 3 Drag vertices of triangle ABC. Look for a relationship between BCe and CeD.[br][br]Step 4 Make a table of at least FIVE distance measurements on a separate piece of paper.[br][br]Step 5 Move a vertex of the triangle to enter the distance measurements into your table.[br][br]Investigate[br][br]1. Keep changing the triangle and adding entries to your table until you find a relationship between the distances BCe and CeD that holds for any triangle. Complete the Centroid Conjecture in your conjecture packet.