GEOMETER'S SKETCHPAD
ASSIGNMENT #7 (Youngberg)
Note the following instructions as you complete this assignment and all other GSP assignments:
•
•
•
Although you will be emailing your assignment to me, save a copy of all of your sketches to your disk.
Answer all of the questions posed to you in a text box at the bottom of the sketch you are currently working
on.
Make sure that all of your sketches are constructions, not drawings. That is, create all sketches so that the
required geometric relationships are maintained when any object is “dragged.”
1. In a new sketch, construct an arbitrary triangle, ∆ABC.
Construct the three perpendicular bisectors of ∆ABC. Recall that they are concurrent at a point called the
circumcenter of the triangle. (We saw this in the last assignment.) Label the circumcenter of your triangle O.
Hide everything except ∆ABC and the point O.
Construct the three angle bisectors of ∆ABC. Recall that they are concurrent at a point called the incenter of the
triangle. Label the incenter of your triangle I. Hide everything except ∆ABC and the points O and I.
Construct the three medians of ∆ABC (see page 857 of your text for the definition of a median of a triangle). The
medians are concurrent at a point called the centroid of the triangle. Label the centroid of your triangle G. Hide
everything except ∆ABC and the points O, I and G.
An altitude of a triangle is a line through the vertex of the triangle that is perpendicular to the opposite side of the
triangle. Construct the three altitudes of ∆ABC. They are concurrent at a point called the orthocenter of the
triangle. Label the orthocenter of your triangle H. Hide everything except ∆ABC and the points O, I, G and H.
Drag your vertices of your triangle and observe how O, I, G and H move in relationship to each other.
Drag A, B, and C so that O, I, G and H all land on top of each other. Make a conjecture about what kind of
triangle ∆ABC must be in order for all four centers to coincide.
QUESTION 1: State your conjecture.
Drag the vertices of your triangle again so that they no longer lie on top of each other. Again, observe how the
centers move in relationship to one another.
QUESTION 2: Which three of O, I, G and H are always collinear? (Even as vertices of the triangle are dragged.)
The line going through these three points is called the Euler line. Construct the Euler line.
Measure the distance between O and G and the distance between G and H. Make a conjecture about the
relationship between the lengths of segment OG and segment GH. Move the vertices of your triangle around to
be sure that the relationship you observed always holds.
QUESTION 3: State your conjecture.
Save your sketch as GSP 7A (triangle centers).
2. In 1820, French mathematicians Charles Brianchon (1785 –1864) and Jean Victor Poncelet (1788-1867)
published a paper that contained a proof of the following statement: “The circle that passes through the feet of the
perpendiculars, dropped from the vertices of any triangle on the sides opposite them, passes also through the
midpoints of these sides and through the midpoints of the segments that join the vertices to the point of
intersection of the perpendiculars.”
Did you catch all that? This circle is called the nine-point circle. Agreed, this statement is somewhat difficult to
follow, but perhaps it will have more meaning for you if you actually construct a nine-point circle.
Construct a large arbitrary triangle. Label it ABC. Using a dashed lines (not segments), construct the three
altitudes of the triangle and label the points where the altitudes intersect the triangle P1, P2 and P3. Construct
the orthocenter and label it H.
Construct the midpoint of each side of the triangle. Label these points P4, P5, and P6 so that P4 is the
midpoint of segment AB, P5 is the midpoint of segment BC, and P6 is the midpoint of segment CA.
Construct the midpoints of segments AH, BH, and CH and label the points P7, P8, and P9 so that P7 is the
midpoint of AH, P8 is the midpoint of BH and P9 is the midpoint of CH. (Hint: You first have to construct
the segments AH, BH and CH before you can construct their midpoints.)
Construct the line segments connecting point P4 to P9, P5 to P7, and P6 to P8. They should all intersect at one
point. Label that point O.
Construct the circle with center O that goes through the point P1. It should pass through all nine points: P1,
P2, P3, P4, P5, P6, P7, P8 and P9. Try dragging any of the vertices of the triangle. If you have constructed the
circle correctly, all nine points will stay on the circle no matter which vertex is dragged. This is the nine-point
circle. (Pretty cool, huh?)
Finally, hide everything but the circle, the triangle, and the points P1 – P9 so that your sketch is less cluttered.
Your resulting sketch should look something like this:
B
P8
P2
P1
A
P4
P5
P7
P9
P3
P6
C
Save your sketch as GSP 7B (nine point circle).
3. Do problem #31, parts (a) and (b), on page 829 of your text. Save your sketch as GSP 7C (angles in a circle).
4. Do problem #32, parts (a) and (b), on page 830 of your text. Save your sketch as GSP 7D (inscribed
quadrilateral).
5. Do the attached handout: “Constructing a Sketchpad Kaleidoscope.” Save your sketch as GSP 7E
(kaleidescope).