Transcript Document

Voila! Proofs With Iteratively
Inscribed Similar Triangles
Christopher Thron
Texas A&M University – Central Texas
[email protected]
www.tarleton.edu/faculty/thron
Why’s it so great to iterate?
• Ancient: “method of exhaustion” was
used by Archimedes to find areas.
• Modern: Fractals (now part of the
standard high school geometry
curriculum)
• Visually appealing, and amenable to
modern software.
• Hugely important technique in modern
analysis
• Can lead to proofs that are visually
immediate (“Voila!”)*
* Although technical details can be nasty
Archimedes updated: parabola section =4/3
1
2
Skew transformation:
Original parabola
section: tangent at
 vertex is || to
base
3
Break up 3 ’s:
Evaluate area: (1 + ¼) 
original 
doesn’t change areas:
parabola parabola
4
Iterate: perform skew
transformations on
each
:
1 + ¼(1+¼ (1+¼ (…
…))))  4/3
1. The Centroid Theorem
• The three medians of a triangle meet at a
single point (called the centroid)
• The centroid divides each median in the ratio
of 2:1
Puzzle pieces:
Assemble:
What does this (appear to) show?
 Blue, red, and green lines all meet at a
single point.
 Dark-colored segments are 2 as long
as light-colored segments
Filling in the details:
A. Why do the triangles fit in the holes? (and can you prove they do?)
B. Why do the same-colored segments line up?
C. How do we know that the segments all meet at a point?
A. SAS similarity, SSS similarity
B. Corresponding segments in similar pieces are ||
’s flipped by 1800 still have corresponding segments ||.
C. Completeness property --Cauchy seq. in the plane converges to a unique point.
Closure property: a line in a plane contains all its limit points.
Summary:
½
1
3
2
½
Details:
4…
SAS & SSS similarity (to get central  to fit)
Corresponding ’s of || lines (converse)
½  & 180o flip preserves ||
Unique || line through a given point
Completeness of plane: and closure of line
½
2. The Euler Segment
• The circumcenter, centroid, and orthocenter
of a triangle are collinear
• The centroid divides the segment from
orthocenter to circumcenter in the ratio 2:1.
What does this (appear to) show?
The points all lie on a single segment
The line must contain the centroid,
because the triangles shrink down to the
centroid.
By considering lengths of segments, the
centroid splits the segment as 2:1
Filling in the details:
A. Why do the points all lie on the same line?
B. Why is the centroid on the line?
C. Why is the |Ortho-Centroid|: |Circum – Centroid| = 2:1?
A. Similar reasoning to last time: unique parallel line through a given point
B. Cauchy sequence, completeness, closure of line
C. Lengths are obtained as alternating +/- sum of segment lengths
Summary:
½
1
3
2
½
4…
½
 The above example iterates the operation of inscribing
180o-rotated similar ’s.
 Try inscribing similar ’s at other ’s 180o.
 Depending on , there are three cases:
Clockwise
Counterclockwise
Figures drawn with: C.a.R. (Compass and Ruler),
zirkel.sourceforge.net/JavaWebStart/zirkel.jnlp
Inverted
Given ABC (clockwise). Successively inscribe similar ’s at any clockwise angle .
The inscribed ’s converge to a point P with the property: PAB = PBC = PCA.
Summary:
• The “equal angle” point P is unique (Proof: “3 impossible regions”)
• P is called a Brocard point
• Any sequence of clockwise-inscribed similar ’s will converge to the
Brocard point, as long as the  size 0 (the 3 fan shapes are always
similar)
• The vertices of the fan shapes lie on three logarithmic spirals: of the
form: ln(r) = k + Cj,: