Lab #3 Assignment
M 333 L Spring 2017
Sketchpad Assignment (4 sketches)
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The nine (9) special points on the Nine-Point Circle of  ABC
FA , FB , FC = the feet of the altitudes from vertices A, B, and C to the sides (perhaps extended) of
the triangle opposite vertices A, B, and C, respectively.
M A , MB , MC
QA , QB , QC
= the midpoints of the sides of the triangle opposite vertices A, B, and C, resp.
= the midpoints of the segments connecting the orthocenter H to vertices A, B, C, resp.
Note: The color specifications are only for your sketches as they appear on your computer monitor. If
a color printer is not available, then black and white printouts are acceptable.
The Sketchpad Assignment is to perform the following steps:
(Note: Only those objects assigned to be constructed should remain visible. Any other points and
lines constructed in the process of constructing the assigned objects must be hidden in the final sketch.)
1) In a new sketch, construct three (3) free non-collinear points and label them A, B and C.
2) Construct three (3) dashed blue complete lines, each passing through two of the three points; that is,
 
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construct AB , BC , and AC .
3) Construct three (3) dashed red complete lines, each passing through one of the three points and
perpendicular to the blue dashed line passing through the other two points.
4) Construct the points which are the feet of the altitudes of  ABC and label them FA , FB and FC .
(Review the Lab #1 Assignment to recall how to put subscripts in a label.)
5) Construct three (3) thick blue segments on the sides of  ABC; that is, construct AB , BC, and AC .
6) Construct the midpoints of these three thick blue segments, and label as MA the midpoint of the side
of Δ ABC opposite A, label as MB the midpoint of the side of Δ ABC opposite B, and label as MC the
midpoint of the side of  ABC opposite C.
7) Construct the orthocenter of  ABC and label the orthocenter point H. (Note: no new lines need to
be constructed. You only need to construct the point of intersection of two previously constructed lines.)
8) Construct the circumcenter and label the circumcenter point S. [ Note: any new lines (such as the
perpendicular bisectors of the sides of the triangle) constructed in the process of locating the
circumcenter must be hidden after the circumcenter has been constructed. ]
9) Construct the segment HS ; construct its midpoint and label the midpoint point N; and then hide the
segment HS leaving the point N visible.
10) Construct QA , QB and QC as follows:
Construct segment AH ; construct its midpoint and label it QA ; then hide segment AH ;
Construct segment BH ; construct its midpoint and label it QB ; then hide segment BH ;
Construct segment CH ; construct its midpoint and label it QC ; then hide segment CH .
11) Construct segment AS ; construct its midpoint and label the midpoint T; construct segment ST ;
construct a circle centered at point N with radius equal to ST (To do this, select together the point N
and the segment ST and then select the option “Circle by Center and Radius” in the “Construct”
menu.); finally, hide segments AS and ST , and hide point T, and make the circle red and thin.
12) If all the previous steps have been performed correctly, you have constructed the Nine-point Circle
of  ABC . Check to make sure that all nine (9) special points lie on the circle.
13) Construct the Centroid by constructing the point of intersection of two medians and label the
centroid point G; hide the medians used in the construction of centroid G.
14) Construct a complete line passing through H and S; make the line green and thin and label the line
“Euler Line”. (Check to make sure that N and G also lie on this line.)
15) Measure the distances HS, HN, HG, NMA , NQA , NFA , and SA.
16) Calculate the ratios: SA / NMA , HN / HS , HG / HS . (Use the Sketchpad calculator in the
Measure Menu and click on a distance to use it in a calculation.)
17) Drag the vertices A, B, and C around and see which measurements and calculations change and
which measurements and calculations remain the same.
18) Identify to yourself the calculations which do not change and verify for yourself that the values of
the non-changing calculations are indeed the numbers that the theory we are studying predicts they
should be.
19) Use “Save as…” to save the sketch in three different files named “lab3_01.gsp”, “lab3_02.gsp”, and
“lab3_03.gsp”.
20) Open file “lab3_01.gsp”; position the vertices A, B, and C so that Δ ABC is an acute triangle which
is not an isosceles triangle; position the labels so that the labels of the nine special points on the ninepoint circle are outside the circle;
IN A CAPTION, identify the sketch as “Lab #3, Sketch #1” and then write “The Nine-Point
Circle of an Acute Triangle”.
Save the file and print the sketch to be handed-in.
Remember to use “Fit to Page” in the Print Preview first.
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21) Open file “lab3_02.gsp”; position the vertices A, B, and C so that  ABC is an obtuse triangle with
the obtuse angle at vertex A; position the labels so that the labels of the nine special points on the ninepoint circle are outside the circle; IN A CAPTION, identify the sketch as “Lab #3, Sketch #2” and then
write “The Nine-Point Circle of an Obtuse Triangle”. IN A SECOND CAPTION, write the following
filling in the blanks:
“The Orthocenter and the Circumcenter are __________(inside or outside) the obtuse triangle and
they lie ___________ (“on the same side of” or “on opposite sides of”) of the triangle segment opposite
the obtuse angle.”
Save the file and print the sketch to be handed-in.
22) Open file “lab3_03.gsp”; measure  BAC and then position the vertices A, B, and C so that Δ ABC
is a right triangle with the right angle at vertex A; position the labels so that the labels of the nine
special points on the nine-point circle are outside the circle; IN A CAPTION, identify the sketch as
“Lab #3, Sketch #3” and then write “The Nine-Point Circle of a Right Triangle”. IN A SECOND
CAPTION, write the following filling in the blanks:
“In a right triangle, the nine special points reduce to _________ points because some of them
coincide.”
Save the file and print the sketch to be handed-in.
23) Click on “File” and select “New Sketch”. Use “File / Save as…” to save as file “lab3_04.gsp”;
In the blank sketch, construct an equilateral triangle  ABC as follows:
a) Construct two points , A and B, and construct the segment AB .
b) Select the point A and, in the Transform Menu, select “Mark Center”.
(Alternatively, you can double-click on point A so that it swells out and back.)
c) Select point A, point B, and the segment AB .
d) In the Transform Menu, select “Rotate” and type “60” in the degrees field of the Rotate Window
and then click on “Rotate”.
e) Label the new point created “C”.
f) Select the point B and, in the Transform Menu, select “Mark Center”.
(Alternatively, you can double-click on point B so that it swells out and back.)
g) Select segment AB and in the Transform Menu, select “Rotate” and type “–60 ” in the degrees
field of the window and then click on “Rotate”.
An equilateral triangle  ABC should now have been completely created.
Perform steps 2) through 13) above using the vertices A, B and C of this sketch, which produces an
equilateral triangle. If any point to be constructed already exists, then do not construct it anew.
Measure all three angles,  BAC,  ABC  BCA . Position the labels so that the labels of the nine
special points on the nine-point circle are outside the circle. To add multiple labels for points which
coincide, use a small caption with the second label. IN A CAPTION, identify the sketch as “Lab #3,
Sketch #4” and then write “The Nine-Point Circle of an Equilateral Triangle”. IN A SECOND
CAPTION, write the following filling in the blanks:
“In an equilateral triangle, the nine special points reduce to _________ points because some of them
coincide. Also, five special triangle points, the _________, the ____________, the ______________, the
_____________, and the _______________ all coincide at the center of the Incircle. Finally, the Ninepoint circle and the ____________ (name a specific circle related to the triangle) coincide.”
Save the file and print the sketch to be handed-in.
Again note: The color specifications are only for your sketches as they appear on your computer
monitor. If a color printer is not available, then black and white printouts are acceptable.
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