Special Segments in a Triangle (pick a triangle, any triangle)

Perpendicular Bisectors of the sides of a triangle: 1. Draw a triangle 2. Construct the perpendicular bisector of each side. 3. The point where all 3 meet is called the “Point of Concurrency” 4. The name of the point of concurrency of the perpendicular bisectors of the sides is: the Circumcenter. It is equidistant from all 3 vertices of the triangle! 5. It’s called the circumcenter because it is the center of the circle that intersects the triangle in only the 3 vertices, i.e., it goes around the triangle and only hits the 3 vertices. It is equidistant from the vertices. P ┐ ┐ ┐

Angle Bisectors of a triangle : 1. Draw a triangle 2. Construct the angle bisector of each angle. 3. The name of the point of concurrency of the three angle bisectors is: the Incenter. It is equidistant from all 3 sides (distance is measured perpendicular to the sides, the shortest possible). 4. The three radii are congruent. The Incenter is equidistant from the sides of the triangle.

Medians of a triangle (segments drawn from a vertex to the midpoint of the opposite side): 1. Draw a triangle 3. Construct each median (see above for def of median). 2. Locate the midpoint of each side. 4. The point of concurrency is called the centroid, or the center of gravity. The triangle will balance if a pencil point is carefully placed on the centroid. 5. The centroid is located 2/3 of the distance from the vertex to the midpoint of a side, no matter which median you choose. M C A 6. So if AM=9, AC would = 6 and CM= Another way to think of this is that the distance of the centroid to the vertex is twice as much as the distance of the centroid to the midpoint!

Altitudes of a triangle: Segments from vertex that are perpendicular to the opposite side. 1. Draw a triangle 2. Construct a segment that starts at a vertex and is perpendicular to the opposite side. 3. Repeat for the other 2 altitudes ┐ ┐ ┐ 4. The point of concurrency is called the orthocenter. Unlike the other three points of concurrency, the orthocenter has no special properties.

Surprises! 1. If you join the endpoints of the altitudes drawn in the previous slide (the orthocenter), you will form a new smaller triangle. The orthocenter of the original triangle will be the circumcenter of the new triangle! Try it!! 2. AND, if you draws lines outside the original triangle that pass through a vertex and are parallel to the opposite side, the orthocenter of the original smaller triangle is the incenter of the new, larger triangle. A B C KL M 3. The orthocenter of ΔABC is the incenter of ΔKLM. Try it!

Questions: 1.Can the circumcenter be inside the triangle? If so, what kind of triangle must it be? Draw an example. 2.Can the circumcenter be on the triangle? If so, what kind of triangle must it be? Draw an example. 3.Can the incenter be outside the triangle? If so, what kind of triangle must it be? Draw an example. 4.Can the incenter be on the triangle? If so, what kind of triangle must it be? Draw an example. 5.Can the centroid be outside the triangle? If so, what kind of triangle must it be? Draw an example. 6.Can the centroid be on the triangle? If so, what kind of triangle must it be? Draw an example.