1. Perpendicular bisectors and angle bisectors within triangles
5-3 Concurrent Lines, Medians, and Altitudes: Part 1 – Circumcenter & Incenter
Perpendicular bisectors and angle bisectors within triangles

2. More paper triangle folding!
Fold an acute triangle so you create a perpendicular bisector of each side.
What do you notice about the folds created by the perpendicular bisectors?
What rule does it suggest about triangle side perpendicular bisectors?
If you use a right triangle and/or an obtuse triangle, does your rule still work?

3. More paper triangle folding!
Fold together sides of an acute triangle so each fold bisects one of the angles.
What do you see about the placement of the folds?
What rule does it suggest about triangle angle bisectors?
If you use an obtuse triangle and a right triangle, does your rule still work?

4. Concurrent lines
Three or more lines that intersect at one point are called concurrent.
The point at which they intersect is the point of concurrency.
In any triangle, 4 different sets of lines are concurrent.
Perpendicular bisectors
Angle bisectors
Medians
Altitudes

5. Theorem 5-6: Circumcenter Theorem
The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices.

6. Theorem 5-6: Circumcenter Theorem
The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter of the triangle.
Points Q, R, and S are equidistant from C, the circumcenter. The circle is circumscribed about the triangle.

7. Theorem 5-7: Incenter Theorem
The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides.

8. Theorem 5-7: Incenter Theorem
The point of concurrency formed by the angle bisectors of the triangle is called the incenter of the triangle.
Points F, E, and G are equidistant from D, the incenter. The circle is inscribed in the triangle.