Chapter 6 Investigation

Using some of the construction methods used in previous sections and definitions covered we will explore the relationship between the altitudes, perpendicular bisectors, medians, and the angle bisectores of a given triangle.

Given an acute scalene triangle

Locate and label the following points on the triangle

 Description of Points

Label of Points
 midpoints of the three sides (3)

E, F, G
 circumcenter (1)

J
 centroid (1)

M
 intersection points of altitudes (3)

N, P, Q
 orthocenter (1)

 S
 midpoints of segments from orthocenter to each vertex (3)

U, V, W
 incenter (1)

X
 midpoint of segment joining circumcenter and orthocenter

Z


Step by Step Instruction

We start with the given triangle

 

a. Using earlier construction methods locate the midpoints of the side of our triangle and label them points E, F, and G

b. Now locate the circumcenter of the triangle. This is the point where the perpendicular bisectors intersect. Therefore, we must first construct the perpendicular bisectors of the given triangle.


Locate the point of intersection and label it point J; then hide the lines to lesson confusion.

 

b. Let's locate the centroid. This is the point where the medians intersect; we'll label this point M. First we must construct the medians of the triangle using previous construction methods.


Now identify the point of intersection of the medians by labeling it point M, then hide the medians


c. Next construct the altitudes of the triangle and identify the points where they intersect the sides of the triangle. Label the points of intersection points N, P and Q.

 

 

 

d. We can now locate the orthocenter, this is the point where the three altitudes intersect. Locate that point on our previous construction and label it point S. Then we can hide the altitudes.

e. Next we want to find the midpoints of segments from the orthocenter to each vertex. To start we must first construct the three segments from the vertexes to the orthocenter.

Now construct the midpoints of the segments and label them points U, V, and W then hide the segments.

 

 

f. Now we will construct the incenter, this is the point where the angle bisectors intersect. First construct the angle bisectors of the triangle.

Locate the point where the angle bisectors intersect and label this point X, then hide the angle bisectors.

e. Now let's locate the midpoint of segment joining circumcenter and orthocenter. First construct a line segment from the circumcenter, point J, to the orthocenter, point S.

This segment is referred to as the Euler line. Then locate the midpoint of this segment and label it point Z

Now that we have located and labeled all 13 points from the chart. Lets investigate them a little further.

First construct a circle whose center is point Z and one of the midpoints of a side of our triangle being another point that lies on the circle.

Looking at our circle how many of our 14 points lie on this circle? Nine of our fourteen points lie on the circle. That is why we call this a nine-point circle, and if you take a close look at our Euler line you will see that four points lie on this line. Leaving only the incenter as not lying on either the circle or the Euler line.

The question is does this remain the case with all triangles, investigate this a little further by doing the same constructions with an obtuse scalene triangle, a right triangle, and an equilateral triangle.

Obtuse Scalene Triangle

Right Triangle

Equilateral Triangle


What assessments can you make about these investigations??


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