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Why Half Angles?

Half angles are a fundamental property of circular geometry. They have been studied since time immemorial, and discussed by many geometricians.

 

The easiest way to describe a half-angle is to show it in the following diagram.

 

The semicircle DEF contains three triangles of interest.

First, is the right triangle. Regardless of where B may fall on DEF, it will always produce a right triangle, DBF.

Second is the right angle produced by ABC. A line CB is erected perpendicular to DF that just touches the circle at B. The hypotenuse AB defines this triangle. This triangle may or may not be a Pythagorean Triangle.

Third is the right angle produced by DBC. The same perpendicular line, CB, is used as one side of this third triangle. The hypotenuse DB defines this triangle. This triangle may or may not be a Pythagorean Triangle.

The angle, BDC, is always one-half the angle, BAC, regardless of where B may fall. I now proceed to demonstrate this fact.

Thales' Theorem

In geometry, Thales' theorem (named after Thales of Miletus) states that if D, B and F are points on a circle where the line DF is a diameter of the circle, then the angle DBF is aright angle.

Proof

We use the following facts: (a) the sum of the angles in a triangle is equal to 180, (b) the base angles of an isosceles triangle are equal, and (c) the diameter of a circle expressed in angular measure is 180.

Let A be the center of the circle. Since AD = AB = AF, ABF and ABD are the two isosceles triangles. (Each has two sides equal, and the two equal sides of each are equal to each other. However, their angles differ.)

(Equation 1):

These conditions make the angles of the isosceles triangle ABF = b + 2a = 180, where b is the angle at the center (in this case) and a is one of the two equal angles.

(Equation 2):

Likewise for the isosceles triangle ADB, d + 2g = 180.

Note that d = (180 - b). This takes advantage of the fact that the diameter of a circle in angular measure is 180.

Then using the alternate expression for d, (180 - b), and substituting into Equation 2, we obtain

(180 - b) + 2g = 180.

Simplifying we find that b = 2g.

Substituting back into Equation 1 we see that 2g + 2a = 180.

Dividing by 2:  g + a = 90.

Q.E.D.

Corollary 1

Equation 2 above shows that b = 2g. This means that g is the half-angle of b. This is true regardless of where B may lie on the circle.

Corollary 2

From Equation 1 we know that 2a  = (180  - b). But (180  - b)  is the alternate angle to b. Hence, a  = (180  - b)/2 = d/2. This means that a is the half-angle of d. This is true regardless of where B may lie on the circle.

Converse

I offer the following without proof.

The converse of Thales' theorem is also true. It states that if you have a right triangle and construct a circle with the triangle's hypotenuse as diameter, then the third vertex of the triangle will lie on the circle.

The theorem and its converse can be expressed as follows:

The center of the circumcircle of a triangle lies on one of the triangle's sides if and only if the triangle is a right triangle.

The circumcircle is that circle that totally encloses the triangle. For a right triangle the diameter is the hypotenuse.

History

Thales was not the first to discover this theorem since the Egyptians and Babylonians must have known of this relationship. However surviving evidence does not provide evidence that they could prove the theorem, and the theorem is named after Thales because he was said to have been the first to prove the theorem, using his own results that the base angles of an isosceles triangle are equal, and that the sum of angles in a triangle is equal to 180

As I have shown, the knowledge of the ancients was far more than empirical. Through the remnants of the mathematical documents available to us we know their level of knowledge was far greater than indicated in the Greek evidence.

Greek Proof

The Greek mathematician, Euclid of Alexandria, (c 325 - c 265 BC), in his Proposition 20, Book 3, gave a general proof that the half-angle statement is true. (Euclid labeled his points differently.)

 

The numbers 1.5 and 1.32 refer to other Euclid Propositions which serve to prove the missing part, and which you may find at Euclid Prop attached.

 

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