Points of Concurrency Lessons 5.2-5.4 Prepared by Mrs. Pullo for her Geometry classes

Concurrent Lines: Circumcenter, Incenter, Centroid, Orthocenter When two lines intersect at one point, we say that the lines are intersecting. The point at which they intersect is the point of intersection. (nothing new right?) Well, if three or more lines intersect, we say that the lines Are concurrent. The point at which these lines intersect Is called the point of concurrency.

The perpendicular bisectors of the sides of a triangle are Theorem 5-6 The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices called the Circumcenter Midpoint is found by adding the x coordinates and dividing by 2, and y coordinates and dividing by 2. To find the perpendicular Line, must find slope and take the negative reciprocal Point of concurrency – CIRCUMCENTER

The Circumcenter is equidistant from all three vertices. Each line is perpendicular to the side and also bisects that side. The Perpendicular Bisector does not have to come from vertex!!

The Circumcenter gets its name from the fact that it is the center of the circle that circumscribes the triangle. Circumscribe means to be drawn around by touching as many points as possible.

So, to find the center of a circle that will circumscribe any given triangle, you need to find the point of concurrency of the three perpendicular bisectors of the triangle. Sometimes this will be inside the triangle, sometimes it will be on the triangle, and sometimes it will be outside of the triangle! Acute Right Obtuse

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

The point of concurrency of the three angle bisectors is another center of a triangle known as the Incenter. It is equidistant from the sides of the triangle, and gets its name from the fact that it is the center of the circle that is inscribed within the circle.

Median: A median of a triangle is the segment That connects a vertex to the midpoint of the Opposite side.

This point of concurrency of the Medians is another center of a triangle called the CENTROID.

From the Centroid to the vertex is 2/3 the length of the median and from the Centroid to the edge is 1/3 the length of the median.

So, if you know the length of any median, you know where the three medians are concurrent. It would be at the point that is 2/3 the length of the median from the vertex it originated from. Theorem 5-8

The Centroid is also the Center of Gravity of a triangle which means it is the point where a triangular shape will Balance.

Altitudes : Altitudes of a triangle are the perpendicular segments from the vertices to the line containing the opposite side. Unlike medians, and angle bisectors that are always inside a triangle, altitudes can be inside, on or outside the triangle.

Theorem 5-9 This point of concurrency of the altitudes Of a triangle form another center of triangles. This center is known as the Orthocenter.

In Conclusion: There are many centers of Triangles. We have only looked at 4: Circumcenter: Where the perpendicular bisectors meet Incenter: Where the angle bisectors meet Centroid: Where the medians meet Orthocenter: Where the altitudes meet.

CONCURRENT LINES Geometry Honors