POINTS OF CONCURRENCY In this lesson we will define what a point of concurrency is. Then we will look at 4 points of concurrency in triangles. As you go through the powerpoint, you will complete your notesheet. You will need to be able to define the 4 points of concurrency and identify them in a picture. You will also use the definition to identify relationships between Segments and angles to solve problems.

POINTS OF CONCURRENCY 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.

Perpendicular bisectors=Circumcenter The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. Point of concurrency

Perpendicular bisectors=Circumcenter This point of concurrency has a special name. It is known as the circumcenter of a triangle. Circumcenter

Perpendicular bisectors=Circumcenter The circumcenter is equidistant from all three vertices.

Perpendicular bisectors=Circumcenter The circumcenter gets its name because it is the center of the circle that circumscribes the triangle. Circumscribe means to be drawn around by touching as many points as possible.

Perpendicular bisectors=Circumcenter Sometimes the circumcenter will be inside the triangle, sometimes it will be on the triangle, and sometimes it will be outside of the triangle! Acute Right Obtuse

Angle bisectors=Incenter The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides. Angle bisector

Angle bisectors=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. Incenter

Medians=Centroid Median: A median of a triangle is the segment That connects a vertex to the midpoint of the opposite side.

Medians=Centroid This point of concurrency of the medians is another center of a triangle. It is known as the Centroid. Centroid

Medians=Centroid This Centroid is also the Center of Gravity of a triangle which means it is the point where a triangular shape will balance.

Centroid Centroid Centroid Theorem The point of concurrency of the medians of a triangle. Centroid Theorem The medians of a triangle intersect at a point called the centroid that is two thirds of the distance from each vertex to the midpoint of the opposite side.

Centroid… PK = 5 Find AP 10 x 2x BP = 12 Find PL. 6 JC = 15 PK + AP = AK JC = 15 Find JP. 5 PK + 2(PK) = AK

In triangle ABC, Q is the centroid and BE = 9 Find BQ Find QE BQ = 2(QE) = 3 OR 6 = 2(QE) 3 = QE

In triangle ABC, Q is the centroid and FC = 15 Find FQ Find QC = 5 QC = 2(FQ) QC = 10 QC = 2(5)

Altitudes=Orthocenter 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.

Altitudes=Orthocenter This point of concurrency of the altitudes of a triangle form another center of triangles. This center is known as the Orthocenter. The Orthocenter of a triangle can be inside on or outside of the triangle.

Find the orthocenter The vertices of triangle FGH are F(-2, 4), G(4,4), and H(1, -2). Find the coordinates of the orthocenter of triangle FGH. Graph the points. Cont…

Find an equation from F to GH. Slope of GH. m = 2 New equation is perpendicular to segment GH. Point F (-2, 4) m = -1/2 y = mx + b 3 = b Cont…

Find an equation from G to FH. Slope of segment FH m = -2 New equation is perpendicular to segment FH. Point G (4, 4) m = 1/2 2 = b Cont…

The orthocenter can be found at the intersection of our 2 new equations. How can we find the orthocenter? If the orthocenter lies on an exact point of the graph use the graph to name. If it does not lie on a point use systems of equations to find the orthocenter. System of equations: Cont…

By substitution. Orthocenter (1, 2.5) 1 = x y = 2.5

POINTS OF CONCURRENCY There are many Points of Concurrency in Triangle. We have only looked at four: Circumcenter: Where the perpendicular bisectors meet Incenter: Where the angle bisectors meet Centroid: Where the medians meet Orthocenter: Where the altitudes meet.

Practice Problems On the half sheet of paper that you were given complete the following problems. 1A-C Identify the point of concurrency that is shown in the triangle B C A

Practice Problems Solve for x, y in each triangle using the given information 2. Point G is a centroid AC = 24, AF=15, AE= 3x-6, BF = 3y 3. Point H is an orthocenter <ABE = 25, <BFC = 8x + 10, <BAE= 6y+5

Practice Problems 4. Point W is both a centroid and an orthocenter. Why is it a circumcenter also? 5. If you draw two medians, is that enough to determine where the centroid of a triangle is? Why/Why not? 6. Do the following matching. Circumcenter W. Angle bisectors Incenter X. Medians Orthocenter Y. Perpendicular bisectors Centroid Z. Altitudes