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Part I - THALES & THE INSCRIBED ANGLE
Thales, known as the Father of Geometry, discovered many of the key concepts about geometry that the Greeks explored for many years after his death and that we rely on today as key foundations to geometrical constructions. Two examples of his discoveries were that a line through the center of a circle bisects the circle into two equal parts, and that all inscribed angles subtending the same arc are equal. It is the latter discovery which I want to expand on.
Inscribed Angles
DFN: An inscribed angle is any angle formed by the two end points of a chord and another point on the circle. All inscribed angles subtended on the same chord (arc) are equal.
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At Thales 
time this way of great practical use
for ship navigation. Ships would often travel quite close to the shore
(so that they wouldn't fall off the edge of a square world) which caused
problems with rocks, coral, and shallow waters. The navigator of the ship
would chart these areas and then find land marks so that they could avoid
damaging their boats. What the navigator did was to construct a circle through
these two landmarks so that the circumference of the circle was outside
of the troubled waters. Thus when an inscribed angle was formed between
the two landmarks and any point on the circle a constant value for an angle
was formed. This angle was the value that was used to guarantee safe passage
because if the boat's angle to these two landmarks was smaller or equal
to the inscribed angle they were safe but if the boat's angle was larger
than the inscribed angle they were heading into dangerous waters. thus if
a student asks you why do we learn this? You can say so that we can navigate
boats.......
PART II - THE BEST KICKING ANGLE
The football rules in college football were changed a few years ago to have the field goal uprights 5 feet narrower than previously. Many commentators have harped about how much harder it is to kick field goals from the hash marks. Assume the field goal is attempted from the hash marks. At what yard marker does the dicker have maximum angle to the two uprights. Is there any merit to some commentators argument to take a penalty in order to have a "better angle" on the field goal kick?
this question I have created a geometer's sketchpad simulation. Here you can see
where the inscribed angles come in. The thick black line on the back of
the end zone is the field goal uprights. The red lines connected to it represent
the kicking angle from any point on the field. The ball is at the 6 yard
line and the approximate kicking angle from that point is 20.8 degrees.
As the ball moves up and down the hash marks we see the kicking angle increase
and decrease. If we continued further down the field towards the 30 yards
line we would see a continue decrease in the kicking angle, thus an undesirable
direction. As we move the ball down the hash marks towards the end zone
we see the angle steadily increase until it hits a maximum of a 40 degree
angle. Unfortunately this occurs approximately in the end zone about 4 years
from the back of the end zone. This is of no help for a kicker. A kicker
usually stands 7 yards back from the line of scrimmage and the closest line
of scrimmage to the ideal angle would be the goal line. Thus with all these
factors the best place to kick the ball would be the 7 yard line and the
angle at that point would be approximately 20 degrees. Concerning the issue
of taking a penalty to get a better angle is not true, anything further
back would create a smaller kicking angle and thus a harder target to hit.
I will mention that this investigation does not look at the vertical angle
for kicking because each kicker drives the ball at a different angle. A
low trajectory kicker might want to back up a bit to get the ball up and
over the defensive line. I will leave this up to the coach to decide.