Math 487 Lab 11 – Part A (Euler Circle and Dilations)

It will be helpful to have a circumcircle tool available for this part of the lab.  The first parts of the lab are most essential.  Do as much as you can during one hour.  (This should be at least through Experiments 1 and 2.)

Basic Construction

Construct a triangle ABC then construct the Nine-Point Circle figure as discussed in class. Specifically,

• Construct a triangle ABC.
• Construct the midpoints A'B'C' of ABC.
• Construct the 3 altitudes of ABC and label the feet of the altitudes D, E, F.
• Construct the orthocenter H as the intersection of two of the altitudes.
• Construct the circumcircle c of ABC and the circumcenter O of this circle.
• Construct the 3 medians and the centroid G where the medians are concurrent.

Pause for organizing the document

• Now stop and make a Hide/Show Button for the whole figure so far.
• Also, go to Document Options and make a page 2 that is a copy of this page in your sketch file.  Also make a page 3.

Dilation Experiment #1

• On page 1, mark G as center and dilate triangle ABC, circle c and point O using the fixed ratio (–1/2).  Call this dilation g.
• What is the image of ABC?
• Consequently what circle d is the image of c? If P is the image g(O), how is P related to d?
• What is the ratio GO/GP?

Now we dilate some more elements by the same dilation.

• Dilate the altitude lines by the same dilation.  What lines do they become in triangle A'B'C'?  What lines are they in triangle ABC?
• From this, tell what is the image of H by this dilation?
• Tell what is the ratio GH/GO.

Dilation Experiment #2

• On page 2, mark H as center and dilate triangle ABC, circle c and point O using the fixed ratio (+1/2).  Call this dilation h..
• What is the image of ABC? Call these points A'', B'', C''. They are the midpoints of the segments HA, HB, HC.
• Consequently what circle d is the image of c? If P is the image of O, how is P related to d?
• What is the ratio HO/HP?

Nine Point Circle Relationships

• In this figure d is the Nine-Point Circle.  Check that indeed A'A'', B'B'', C'C'' are diameters and that these diameters are, in pairs, diagonals of rectangles.

Euler Line

Consider that from the definition of a dilation with center X, the image of a point Y is Y' and the points X, Y, Y' are collinear.

• Use this fact and the relationships above to conclude that the points O, P, G, H are collinear.

• Given what you know about the ratios, if a ruler is put down on the line so that point O is at O and point P is at 1, where would the points G and H be located?

Other Observations

Let the intersection (besides A) of the altitude AD with circle c be D'.  Explain why h(D') = D and thus why D' is the reflection of D across BC.

References:  Berele-Goldman, 9.3 and 9.4.  Also many sites on the web and many geometry books.

Dilation Experiment #3 (this will be demonstrated in lab)

• On page 3, draw in a new point X and dilate the whole figure with center X and ratio (-1/2).  Observe what the dilated figure looks like – transformed by a point reflection and then a scaling by one-half.
• Construct lines through A and its image, likewise B and C with their images.  These lines will pass through X.
• Drag X around to different positions.  Eventually check that if X moves to coincide with G, then the image triangle becomes the midpoint triangle.
• Draw in a new point Y and again dilate the original figure by ratio (+1/2).  (You may wish to hide the image by the X dilation.)  Then drag Y around so get an idea of what it looks like.  Check that when Y is moved to coincide with H, then the Nine-Point Circle figure again appears.