Download Thales and His Semicircle Theorem Historical Context: Suggested

Thales and His Semicircle Theorem
Historical Context:
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When: ca. 585 B.C.
Where: Greek Ionia
Who: Thales of Miletus
Mathematics focus: Investigation of a useful property of a triangle inscribed in
a semicircle.
Suggested Readings:
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Thales and his contributions to mathematics, astronomy, and philosophy:
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Thales.html
• Early Greek geometry:
http://jwilson.coe.uga.edu/EMAT6680/Greene/EMAT6000/greek%20geom/greekg
eom.html
• NCTM’s Historical Topics for the Mathematics Classroom (1969):
“Demonstrative geometry” (pp. 170-172) and “Early Greek geometry” (pp. 172174).
• Key search words/phrases: Thales, Greek geometry, semicircle, deductive
reasoning, Pythagoras, Euclid
Problem to Explore:
Investigation and determine a useful property of a triangle inscribed in a semicircle.
Why This Problem is Important:
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One of first geometrical theorems possibly supported by deductive reasoning.
Key relationship that supports idea of mean proportional in constructing x .
Problem Solving Experiences:
In addition to his many accolades—First Philosopher, First Mathematician, and one
of the “Seven Sages” of Antiquity—Thales of Miletus was a statesman, an engineer,
a businessman, an astronomer, and a teacher. As part of his claim to fame, Thales
established the idea of demonstrative geometry, shifting scientific thought from
mythos (i.e. understanding the world via traditional stories) to logos (i.e.
understanding the world via reasoning). In fact, Aristotle stated: “To Thales the
primary question was not what do we know, but how do we know it.”
Thales is usually credited with being responsible for five theorems in geometry:
• A diameter bisects a circle.
• Vertical angles are equal.
• Base angles of an isosceles triangle are equal.
• Angles inscribed in a semicircle are right angles.
• A triangle is determined by its base and two base angles (i.e. ASA congruence
criterion).
Because all of Thales’s written works have been lost, the nature and quality of his
proofs of these “theorems” remain unknown. Nonetheless, his theorems provide
some fruitful investigations and opportunities to speculate.
Re-consider his fourth theorem, which suggests the idea of a locus that can be
visualized by implementing these steps using GSP:
• Construct a line segment AB with midpoint O
• Draw a circle C with center point O and radius OA
• Construct a random point on D circle C.
• Construct segments DA and DB, forming angle ADB.
C
D
A
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O
B
Measure angle ADB
Dynamically move point D along the circle.
Watching the angle measure of angle ADB, was Thales correct?
1. Prove Thales’ fourth theorem. Hint: Use his third theorem.
What about the converse of his fourth theorem? That is, is the hypotenuse of a right
triangle's the diameter of its circumcircle?
2. Prove that this converse is true. Can you produce more than one proof?
Extension and Reflection Questions:
Extension 1: Many mathematics historians claim that Thales used his geometrical
skills to determine the distance of a ship from a position on the shore. suggests that
Thales’ method perhaps is the same as that of the Roman surveyor Marcus Junius
Nipsius:
• Your goal is to find the distance from your position on shore (A) to the ship (an
inaccessible point B)
• Facing the ship, turn a right angle to your left, and walk a fixed number of
paces. Plant a staff or stake at this place (C)
• Continue walking in that same direction the same number of paces (D)
• Turn another right angle to your left, and walk straight and away from the
shoreline, looking over your left shoulder at the ship and counting your paces
• When your position (E), the staff (C), and the ship (B) are collinear, stop.
• The number of paces walked away from the shoreline (DE) is equal to the
distance of the ship from the shoreline.
Draw a diagram to illustrate this method, prove why DE = AB, and discuss why it is
possible that Thales could have known this method. Finally, what are difficulties that
could distort the answer if one were to try this method in the real world.
Extension 2: Plutarch (46-120), a Roman philosopher and historian of Greek origin,
pretends that Thales is listening to Niloxenus, another of the “Seven Sages”, talk
about Amasis, King of Egypt:
Although he [Amasis] admired you [Thales] for other things, yet he particularly
liked the manner by which you measured the height of the pyramid without any
trouble or instrument; for, by merely placing a staff at the extremity of the shadow
which the pyramid casts, you formed two triangles by the contact of the sunbeams,
and showed that the height of the pyramid was to the length of the staff in the same
ratio as their respective shadows.
Draw a diagram to illustrate this method and prove why the method works. Why is it
possible that this story by Plutarch is fiction because Thales did not know enough
mathematics?
Open-ended Exploration: Try to prove Thales first theorem: A diameter bisects a
circle. First, decide what the statement means. That is, does it mean that the
diameter divides the circle’s circumference into two arcs that have equal lengths and
are congruent? Or, does it mean that the diameter divides the circle’s interior into two
solid semicircles that have equal areas and are congruent? Once you have
determined a meaning, can you prove the statement?