Geometry has a sacred book: Elements. Author of Elements, Euclid, used only these tools when he discovered his geometry:
A compass and an unmarked ruler… He showed what can or can’t be done with them in geometry.
Dividing a Finite Straight Line into Two Equal Parts
We already know that one can draw a straight line segment between any two points. Let’s say we have a straight line between the points A and B. Euclid found an ingenious method to divide AB into two equal parts using his only two tools. Let’s assume that AB is 6 cm.
This way we will know that Euclid’s method works only if we find two parts that are 3 cm long.
According to Euclid one should take point A and point B as the centers of two equal circles with radius AB:
Euclid says that one should define the intersection points as C and D:
Then he suggests one to connect C to D:
At this point Euclid talks about two outcomes:
- CD is perpendicular to AB.
- CD divides AB into two equal parts.
As seen in the picture CD really divides AB into two equal parts. One can use this method and see that those two lines are perpendicular with a quadrant.
Dividing a Random Angle into Two Equal Parts
Obviously Euclid didn’t stop there and tried to figure out if it was possible to divide any given angle into two equal parts. Let’s consider straight lines AB and BC intersects and form a 90-degree angle:
Euclid says that one should take B (the intersection point) as center and draw a circle with random radius. This circle will intersect AB and BC at two points: D and E.
Now Euclid tells us to use our compass and draw two equal circles that have centers D and E:
As seen above, these circles intersect at two points. Let’s choose the point F and connect it with B:
Euclid claims that the straight line BF divides the angle into two equal parts:
We can see with our quadrant angle is divided into two equal parts of 45 degrees.
One wonders…
Is it possible to use Euclid’s methods and divide any given straight line into three equal parts?
Try to answer the same thing for an angle.
M. Serkan Kalaycıoğlu
kmatematikatolyesi.com 