GSP 3.1 – Taking triangle centers to a whole new level!
Page 1: Review of the old centers
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Draw an arbitrary triangle XYZ and construct the circumcenter (Point C). (hide the perp lines)
Complete the following conjectures/observations we made in our last GSP in text box 1: (write the whole
sentence if it’s fill in the blank)
a) When a triangle is a right triangle the circumcenter is _____________.
b) The circumcenter is inside the triangle when the triangle is ______ and outside the triangle when it is
______.
c) The circumcenter is equidistant from the ____________ of the triangle.
d) Describe how to construct a circle that passes through any 3 non-collinear points.
Construct the incenter (I), centroid (N), and orthocenter(O) of triangle ABC (hide the lines you used to construct
so only the triangle and the three centers are visible.
Complete the following conjectures/observations we made in our last GSP in text box 2: (write the whole
sentence if it’s fill in the blank)
a) The incenter is equidistant from the ___________ of the triangle
b) Describe how to construct the incircle of a triangle.
c) The three centers that are always collinear are _______________________.
Page 2: Copy and paste your triangle XYZ and the 4 centers onto page 2. Hide the Incenter
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Construct the midpoints M, L, and P of the sides of triangle XYZ and construct a circle that contains M, L, and P.
Observe the circle’s connections to the triangle and its relationship to the circumcircle and incircle. This new
circle is called the “Euler circle”. Answer the following in text box 3:
a) Can the Euler circle be the circumcircle? If so, when?
b) Can the Euler circle be in the incircle? If so, when?
c) Is the center of the Euler circle (called the Feurbach point – Label it F) ever coincident with any other
centers? If so, when?
d) Is the Feurbach point ever collinear with any other centers? If so, when?
From point X, construct the median, altitude, and angle bisector. Answer the following in text box 4:
a) Does the Euler circle contain the endpoints of any of these segments? Which ones and in which
scenarios?
b) Construct the midpoints of OX, OY, and OZ. What do you notice? (Use general vocabulary!)
The Euler circle is most commonly called “the nine point circle” because of nine interesting intersection points.
From what you have observed, answer the following in text box 5:
a) Explain the name, “nine-point circle” by indicating what 9 points the name may be referring to.
b) In which types of triangles does the nine-point circle lose some of its 9 points (maybe the intersections
don’t exist, maybe an intersection is coincident with another intersection etc)
Page 3: Copy and Paste your triangle XYZ, along with the centers C, O, N, and F onto page 3
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In our last GSP, we recognized that O, C, and N were collinear and maintain a specific order. We are now going
to delve deeper into the relationship between O, C, and N. Measure the distances between the points (OC, ON,
CN). Answer the following in text box 6:
a) There is a very specific relationship between these three centers and their distances from one another.
Explain the relationship. (Make sure your relationship holds for all triangles).
Side Note: We are now going to do a dilation. We haven’t worked with dilation before, so here is a little snapshot:
Dilation:
To perform a dilation, you must choose a center of dilation (P) and a scale factor k. Then a
point A is dilated to A’ iff
A'P
= k and A, P, and A’ are collinear. If k is negative, then A-P-A’
AP
and if k is positive, then A’-A-P or A-A’-P.
A'
A
P
When a segment or figure is dilated, each point undergoes this same dilation.
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Change the color of the circumcenter (C) so it is different than the color of the orthocenter (O). We are going
to choose point N as the center of dilation by double clicking on N until it explodes. Unclick all points. Now,
select the Circumcenter, and go to Transform: Dilate: and enter in the scale factor as -2:1. Do you see what
happened? (Discuss with partner, you do not need to answer on GSP)
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We’re going to do another dilation with the 9 point circle. Double click point N to make it the center of dilation.
Unclick all points. Now, select the 9 point circle and go to Transform: Dilate: and enter in the scale factor -2:1.
Answer the following in text box 7:
a) What happened to the 9 point circle when you preformed this dilation? Be specific (got bigger isn’t
enough)
b) In text box 6, you discovered a relationship between the distances between the orthocenter,
circumcenter, and centroid. Can you find a similar relationship between the centers and the Feurbach
point?
c) Using these ideas, describe what happens to the 9 point circle if you do a dilation using N as the center
of dilation but use a scale factor of -1/2.
Page 4: Another center? Maybe? (minimum distance to sides)
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Construct an arbitrary triangle RST by finding the intersection of lines RS , ST , and RT . Overlay these lines
with thick segments to show the triangle RST clearly. Create a random point P in the interior of the triangle.
Construct perpendiculars from P to the LINES determined by the sides of RST. Label the intersections of the
perpendicular LINES A, B, and C. Triangle ABC is the pedal triangle for point P.
o Lets imagine that the lines RS , ST , and RT represent roads. You are going to live somewhere within
these roads and then you will build small roads from your house to each of these streets. You want to
minimize the amount of material you need to use, so you are trying to determine where you should
build your house so that the distance to all three roads is a minimum.
PA + PB + PC is the sum of the lengths of the roads you would need to make. Calculate this sum
on GSP. Drag point P around and answer the following in text box 8: Where should you build
your house to minimize the sum of the distances to the roads? (To minimize PA + PB + PC).
o Now, lets change triangle RST so it is equilateral. (Make circle S with radius RS and circle R with radius RS
and then merge point T to the intersection of the circles). Now drag point P.
Describe your observations in text box 9.
Page 5: One more center? Maybe? (minimum distances to vertices)
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Construct an arbitrary triangle ABC and point P inside the triangle. Now, let’s pretend that you are on a deserted
island and you want to build your camp so you have a minimum distance to travel to the vertices to alert
potential rescue ships. This time, we are minimizing the sum AP + PB + CP. Calculate this sum.
o Change ABC into an equilateral triangle (as on page 5). In text box 10, describe where P should be
located in an equilateral triangle.
o Change ABC back into an arbitrary triangle. Move P around to try and minimize AP + BP + CP. The correct
location for P is called the Fermat point. Answer the following in text box 11:
a) Is the Fermat point one of our old centers?
b) Does the Fermat point appear to be collinear with the old centers?
c) Measure ∡APB. ∡BPC and ∡CPA . Make a conjecture about the location of the Fermat point.
d) What happens if m∡A > 120?
e) BONUS – find a method to CONSTRUCT the Fermat point – you may do some research ☺