Points of Concurrency in Triangles Keystone Geometry

Review of Segments in a Triangle There are 4 different special segments in a triangle that create angle and segment relationships. Perpendicular bisector – a segment, ray, line, or plane that is perpendicular to a segment at its midpoint. Angle bisector – a ray that divides an angle into two congruent adjacent angles. Median of a triangle – a segment from a vertex to the midpoint of the opposite side. Altitude of a triangle – perpendicular segment from a vertex to the opposite side or line that contains the opposite side (may have to extend the side of the triangle).

Did you notice? When you construct all three medians, they intersect at a single point. This point is a center of the triangle. When you construct all three altitudes they also intersect at a single point, but a different point from before! This is a different center. The same goes for angle bisectors and perpendicular bisectors. A triangle has many different centers.

Points of Concurrency When three or more lines intersect, they are called concurrent lines. The point where concurrent lines intersect is known as a point of concurrency. There are four different points of concurrency in triangles: Incenter Orthocenter Centroid Circumcenter

Point of Concurrency for Perpendicular Bisectors: The perpendicular bisectors are concurrent at a point called the circumcenter. To find the circumcenter: Bisect each side of the triangle to locate its midpoint. Construct a perpendicular line at each midpoint. The point where they meet is the circumcenter.

Since the circumcenter of the triangle is equidistant from the three vertices of the triangle, it can also be found by circumscribing a circle around the triangle and finding it’s center. The circumcenter can be inside or outside of the triangle. It will depend on the type of triangle (acute, right, obtuse) The circumcenter also creates 3 isosceles triangles within the main triangle.

CONCURRENCY OF ANGLE BISECTORS OF A TRIANGLE The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. If AP, BP, and CP are angle bisectors of ∆ ABC, then PD = PE = PF The point of concurrency is called the incenter. Note: The incenter is always “inside” of the triangle. Note: The incenter is equal distance from all three sides.

VS = VT = VU a2 + b2 = c2 152 + VT2 + 172 225 + VT2 = 289 VT2 = 64 VT = 8 VS = 8 Pythagorean Thm. Substitute known values. Multiply. Subtract 225 from both sides. Take Square Root of both sides. Substitute.

In the diagram, D is the incenter of ∆ABC. Find DF. DE = DF = DG DF = DG DF = 3 Concurrency of Angle Bisectors Substitution

Where the medians meet in An Acute Triangle: The Centroid In the acute triangle ABD, figure C, E and F are the midpoints of the sides of the triangle. The point where all three medians meet is known as the “Centroid”. It is the center of gravity for the triangle. B A D E C F

Concurrency of Medians of a Triangle The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side. This point is called the centroid. The distance from the vertex to the centroid is twice the distance from the centroid to the midpoint. AB = AC + CB If AC is 2/3 of AB, what is CB? CB = 1/3 If AB = 9, what is AC and CB? AC = 6 What do you notice about AC and CB? AC is twice CB Why? Now assume, A is vertex, C is centroid, and B is midpoint of opposite side. Vertex to centroid = 2/3 median CB = 3 Centroid to midpoint = 1/3 median

P is the centroid of ∆ABC. What relationships exist? The dist. from the vertex to the centroid is twice the dist. from centroid to midpoint. AP = 2, PE = 1 BP = 2, PF = 1, CP = 2, PD = 1 The dist. from the centroid to midpoint is half the dist. from the vertex to the centroid. PE = ½ , AP = 1, PF = ½, BP = 1, PD = ½, CP=1 The dist. from the vertex to the centroid is 2/3 the distance of the median. BP = 2/3, BF=1, CP = 2/3, CD 1, AP = 2/3, and AE = 1 The dist. from the centroid to midpoint is 1/3 the distance of the median. PE = 1/3  AE= 1 PF = 1/3  BF= 1 PD = 1/3  CD= 1

BG = 12 + 6 BG = 18 What do I know about DG? What do I know about BG? BG = BD + DG BG = 12 + 6 DG = 6 BG = 18

Concurrency of Altitudes of a Triangle The lines containing the altitudes of a triangle are concurrent. This intersection is called the orthocenter. In a right triangle, the legs are also altitudes. In an obtuse triangle, sides of the triangle and/or the altitudes may have to be extended. Notice obtuse triangle, orthocenter is outside the triangle. Notice right triangle, orthocenter is on the triangle.

If the triangle is obtuse, such as the one on pictured below on the left, then the orthocenter will be exterior to the triangle. If the triangle is acute, then the orthocenter is located in the triangle's interior If the triangle is right, then the orthocenter is located at the vertex of the right angle.