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1. Definition of similarity (transformation) with ratio of similitude
k. Bix, p. 257.
2. Thm. Given a similarity transformation and three points A,
B, C. Let the images by F of these points be A', B', C'. If
ABC is a triangle, then A'B'C' is a similar triangle. If A, B,
C are collinear, then A', B', C' are collinear and AC/AB = A'C'/A'B'.
This implies that lines are mapped to lines. Also the definition
implies that circles are mapped to circles.
3. Definition of dilation with (signed) ratio r and center A.
4. Thales implies any dilation is a similarity transformation.
5. Definition of dilatation. Bix, p. 256.
6. Thales implies that any dilation is a dilatation. Other dilatations
are translations and identity. (Why no others? See Construction
below.)
7. Construction A: Given a center O, a point A and a point
A' on line OA, there is a unique dilation F with center O and
F(A) = A'. For any point B, the point B' = F(B) can be constructed
using parallel lines. (Note special case when B is on line OA.)
8. Construction B: Given a segment AB and a parallel segment
A'B', there is a unique dilatation G with G(A) = A' and G(B) =
B'. For any point C, the point C' = G(C) can be constructed using
parallel lines. (Note special case when line AB = line A'B'.
Also, note that previous construction is a special case of this.
Corollary: there is exactly one dilatation that takes AB to A'B'.)
9. Construction C: Given a segment AB and a parallel segment
A'B', construct center E of the unique dilatation G with G(A)
= A' and G(B) = B' and also the center I of the unique dilatation
H with H(A) = B' and H(B) = A'. (Special case: |AB| = |A'B'|.)
10. Construction D: Given two circles, construct the two
centers of similitude. (Special cases: circles of the same radius
or concentric circles.) Bix, p. 259, also GTO.
11. State and prove the connection between centers of similitude
of two circles and common tangents.
12. Formula for dilations and other dilatations and relationship
to parametrization of a line..
13. Center of a composition of F and G is on the line of centers
of F and G (special case: when F, G or FG is a translation).
14. Menelaus theorem is an example of composition of dilations.
15. What is the locus of midpoints of a segment AB, where A is
a fixed point and B is any point on given circle c?
16. Given a triangle ABC, construct a square DEFG so that DE is
on segment AB, F is on BC and G is on AC (assume neither angle
A or angle B is obtuse).
17. Given angle ABC and a point D inside this angle, construct
all circles through D that is tangent to rays BA and BC.
18. Construction E: Given two circles, construct the common tangent
lines.
19. Given 3 circles, the 6 centers of similitude lie on 4 lines,
with 3 on each line. (There are some special cases.)
20. In a triangle A = triangle A1A2A3, let M1, M2, M3 be the midpoints
of the sides and let G be the centroid. Prove that the dilation
D with center G and ratio 1/2 maps triangle A to triangle
M = triangle M1M2M3.
21. Let O be the circumcenter of triangle A and let T be the circumcenter
of triangle M; prove that D takes O to T.
22. Let H be the orthocenter (the point of concurrence of the
altitudes) of triangle A. Prove that O is the orthocenter or
triangle B. Thus conclude that D takes H to O.
23. Use the previous results to conclude that G, O, H, T are collinear.
The line through these points is called the Euler Line. Compute
the ratios GH/GO and GO/GT. Sketch where the 4 points lie on
a line and the relative distances between the points. In particular,
what is TH/TO?
24. The circumcircle of triangle M is called the Nine-Point Circle
of triangle A. The points G and H are the internal and external
centers of similitude of the circumcircle c of triangle A and
the nine-point circle b of triangle A. (Hint: D takes c to b,
and this shows that G is the internal center. Let R be the half-turn
with center T; show that RD is the other dilation that takes c
to b. Check that RD fixes H, so H is the center.
25. Since RD has ratio 1/2, the midpoints N1 of HA1, N2, of HA2,
and N3 of HA3 are all on b, and in fact M1N1, M2N2 and M3N3 are
diameters of b.
26. The Carpenter Principle applied to b shows that the feet,
F1, F2, F3, of the altitudes of triangle A also lie on this circle
b. The nine points M1, M2, M3, N1, N2, N3, and F1, F2, F3 are
the nine points referred to in the name of b, the nine-point circle
of A.