TOPIC 15
Concurrence in triangles
When two lines meet, we call their common point an intersection. When three of more lines
meet, we often call that common point a point of concurrence. That word means nearly the
same as it does when people are discussing issues or policy. They “concur” when they reach an
agreement, or when their ideas meet.
We will now do an investigation about when lines in triangles concur. We will look at what
different points of concurrence can happen, what they are called, why they might be important to
know about, and how they are related to each other. We will also use the results to do some
calculations, using coordinate geometry.
This topic is rich in investigation and numerical work, more than deductive proof.
Sketchpad Demonstration of Concurrence in Triangles
Use Geometer’s Sketchpad to visualize how various lines behave in triangles. You will be asked
questions after you have completed the sketches. This is an inductive thinking exercise, so use
the capabilities of Sketchpad to change the final drawing so that you have several examples from
which to draw a conclusion.
Note that the words such as select, construct, measure, and label are specific terms in
Sketchpad, and there are tools for each of them (on the left side of the sketch or along the top).
1.
Open a new Sketch.
2.
Construct a triangle with any three segments (Use the Segment Tool). Label the vertices
(with the Text Tool) A, B, and C.
3.
Select one side, and construct the perpendicular bisector of that side. It will be a line.
Select each of the other two sides (one at a time), and construct their perpendicular
bisectors.
Definition: Three or more lines concur if they all intersect at one point.

Do the perpendicular bisectors concur?
Topic 15: Concurrence in Triangles

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If “yes”, where is the point of concurrence with respect to the triangle?
4.
Select two of the perpendicular bisectors, and highlight both of them (Hold the Shift key
down to select more than one item.) Construct the point of intersection. Label it as D.
5.
Construct the line segment connecting D with any one vertex of the triangle.
6.
Select D and the line segment you just constructed. Construct a circle.
7.


The triangle is inscribed in this circle
The circle is circumscribed about the triangle.

Since this is a circumscribed circle, D is called the circumcenter of the triangle.
Grab A, B, or C and move it.

Do the perpendicular bisectors always concur?

Do the perpendicular bisectors create the circumcenter of the triangle in all
instances?

Where can circumcenter of the triangle be with respect to the triangle?
8.
Open a new sketch.
9.
Construct a triangle called ΔDEF.
10.
Select one angle (Make sure that you select it with its three-letter name), and construct
the angle bisector.
11.
The angle bisector is a line. Select the angle bisector and the side it intersects. Construct
the intersection. Label it G.
12.
Select the vertex and G. Construct a line segment.
13.
HIDE the line which is the angle bisector. Keep the angle bisector segment in view.
Topic 15: Concurrence in Triangles
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14.
Repeat the construction with the two other angle bisectors, labeling the intersections as
H and I on the opposite sides.
15.
Construct the point of concurrency. Label it J.
16.
Select J and one side of the triangle. Construct a perpendicular line.
17.
Construct and label the point of intersection of the perpendicular and the opposite sides.
Label it K.
18.
Construct segment JK .
19.
Use J as center and radius JK , and construct a circle.


The circle is inscribed in the triangle.
The triangle is circumscribed about the circle.

Since this is an inscribed circle, J is called the incenter of the triangle.
20.
Grab D, E, or F and move it. Observe where the incenter is for each triangle.
21.
Open a new sketch


22.
23.
Def: A median is a line segment from a vertex to the midpoint of the opposite
dies.
There are three medians in a triangle.
Construct a triangle and its three medians. Call the triangle ΔLMN. If the medians
concur, label the point of concurrence as O.

Do the medians concur?

If so, the point of concurrence is called the centroid of the triangle.
For each median, measure the piece of the median from the vertex to the centroid, and the
piece from the centroid to the opposite side. (To MEASURE, select two points and select
MEASURE/LENGTH or MEASURE/DISTANCE) Record the result from one triangle
below:
Topic 15: Concurrence in Triangles
24.
25.
Vertex to centroid= ______
centroid to opposite side = ______
Vertex to centroid= ______
centroid to opposite side = ______
Vertex to centroid= ______
centroid to opposite side = ______
Move a vertex of the triangle, and measure again.

Look at the pairs of measurements into which the median is separated by the
centroid. What conclusion can be made about where the centroid is located along
a median?

Use one of the cardboard triangles. Draw the medians on the triangle. Put one
finger at that point and see if you can balance the triangle there.
Open a new sketch.

26.
27.
page 4
Def: An altitude of a triangle is a line segment from a vertex which is
perpendicular to the opposite side (or the opposite side extended).
Construct a triangle labeled as PQR. Construct the three altitudes. (Remember that
they are segment, not lines.)

Do the altitudes concur?

If so, the point of concurrence is called the orthocenter. (Note: “Ortho” comes
from “orthogonal” which is almost a synonym for “perpendicular”)
Move the vertices of the triangle around.

Where can the orthocenter be located with respect to the triangle?
Topic 15: Concurrence in Triangles
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
Describe the triangle when the orthocenter is inside the triangle.

Describe the triangle when the orthocenter is inside the triangle.

Describe the triangle when the orthocenter is inside the triangle.
Concurrence of Lines in a Triangle
with Coordinate Geometry
Review of skills:
1
which passes through the point (1 , -3)
2
28.
Write the equation of a line with slope 
29.
Find the distance between (1 , 5) and (-3 , 8)
30.
Where is the midpoint of (6 , 9) and (10 , 21)?
31.
If A is at (4 , 0) and M is at ( 7 , -1), and M is the midpoint of AB , what are the
coordinates of B?
32.
Write the equation of the line which is parallel to 2x – 3y = 12 which passes through (2 ,
6).
Topic 15: Concurrence in Triangles
33.
page 6
Write the equation of the line which is perpendicular to x + 3y = 8 which passes through
(4 , 0).
Use this triangle for
#34 through #44.
B
C
A
Medians in a triangle.
34.
Locate the coordinates of the three midpoints of the sides of ABC .
35.
Write the equation of each median. Label each with the name of its segment.
Topic 15: Concurrence in Triangles
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36.
Find the intersection of any two medians algebraically, using the equations you just built.
37.
Find the intersection of any other two medians algebraically.
38.
Find the length of any one median. Then find the length of the two segments into which
it is divided by the point of concurrence. (Do not assume the conjecture we found a few
days ago. You are going to see if it works with coordinate geometry.)
Topic 15: Concurrence in Triangles
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Perpendicular Bisectors of the sides of a triangle.
39.
Write the equations of each of the perpendicular bisectors of the sides of ABC (above).
A perpendicular bisector passes through a midpoint of a side and is perpendicular to that
side.
40.
Find the intersection of any two perpendicular bisectors algebraically, using the equations
you just built.
41.
Find the intersection of any other two perpendicular bisectors algebraically.
Topic 15: Concurrence in Triangles
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Altitudes in a triangle.
42.
Write the equations of each of the altitudes to the sides of ABC (above). An altitude
passes through a vertex of a triangle and is perpendicular to the opposite side.
43.
Find the intersection of any two altitudes algebraically, using the equations you just built.
44.
Find the intersection of any other two altitudes algebraically.
Topic 15: Concurrence in Triangles
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Concurrence applications
Think about the various kinds of concurrence that you have investigated. Decide how the
significance of them applies to each of these practical situations.
Where are all of the points on an angle bisector?
Consider this theorem. Provide an informal argument.
Given:
Prove:
So we could say that, “All of the points on the bisector of an angle are __________________
from the sides of the angle.”
Where are all of the points on the perpendicular bisector of a line segment?
Consider this theorem, and provide an informal argument.
Given:
Prove:
Topic 15: Concurrence in Triangles
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We could say, “All of the points on the perpendicular bisector of a line segment are
________________________ from the _________________________________.”
Some applications:
45.
Two rivers bound a property, one from A to Y and another from A to X. Water from the
rivers will be used to irrigate the field using a linear, mobile irrigation system (shown
below). If equal amount of water will be taken from each river, and the cost to move that
water is determined by the distance it has to travel, where should the irrigation system be
placed on the property?
http://www.peerlessequipment.com/
Topic 15: Concurrence in Triangles
46.
Locate a lifeguard station to serve the beaches called PQ and PR .
47.
Locate a mall with restaurants to serve the towns located at P and R.
What happens if you also want to serve a town located at Q?
48.
Locate a hospital for villages at R, Q, and L.
page 12
Topic 15: Concurrence in Triangles
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49.
Locate a pizza store for schools located at Q,R,L, F, and S.
50.
Consider the state of Florida. You are going to build a cellular phone hub. The distance
to serve cities needs to be considered for efficiency, so you want to keep the distance
constant to the three corners of the state (i.e. Pensacola, Jacksonville, and Miami).
(Actually minimizing total distance eventually becomes the major issue.) Where should
you locate the cellular hub?
51.
The Sprinkler Problem
The Parks Department is installing a circular sprinkler (a sprinkler whose spray makes a
perfect circle) to water the lawn at a park that is in the shape of a triangle and is
surrounded by sidewalks (see fig. 1). The sprinkler should be placed so as to cover as
Topic 15: Concurrence in Triangles
page 14
much lawn as possible without spraying the sidewalks. Where should the sprinkler be
placed?
What if a second sprinkler is going to cover as much as possible of the lawn on the right hand
side (quadrilateral-ish shape)? Where should the second sprinkler be placed?
from:
http://www.cnseoc.colostate.edu/docs/math/mathactivities/june2007/Sprinklers%20and%20Amusement%20Par
ks.pdf
52.
Topic 15: Concurrence in Triangles
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From
http://www.google.com/search?q=angle+bisectors+applications&nord=1&rlz=1C2LENP_enUS5
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y7TH3P5Ur3tcRM%253A%3B3QnIddZw1eoX5M%3Bhttp%253A%252F%252Fmathworld.wo
lfram.com%252Fimages%252Fepsgif%252FExteriorAngleBisectors_1000.gif%3Bhttp%253A%252F%252Fmathworld.wolfram.co
m%252FExteriorAngleBisector.html%3B397%3B307
We are going to produce the drawing on the previous page. Draw any triangle. Construct the
bisectors of each of the exterior angles of the triangle. At the points where pairs of these exterior
angle bisectors intersect, construct circles tangent to the original triangle. Discuss the size of
these three circles. (Leave plenty of space, as the circles are often large.)
Topic 15: Concurrence in Triangles
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53.
Look up Ceva’s Theorem. See if you can find out what conditions are needed for it to be true.
Read through a proof. Do we know enough know to be able to follow this proof?
Topic 15: Concurrence in Triangles
54. Read through the instructions on this activity.
From: http://gp.lethsd.ab.ca/mrbrunner/PublishingImages/3.3%20Textbook.pdf
page 17