Thales' theorem

In geometry, Thales' theorem (named after Thales of Miletus) states that if A, B and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle.

Proof

We use the following facts:

the sum of the angles in a triangle is equal to two right angles (π),
the base angles of an isosceles triangle are equal.

Let O be the center of the circle. Since OA = OB = OC, OAB and OBC are isosceles triangles, and by the equality of the base angles of an isosceles triangle, OBC = OCB and BAO = ABO. Let γ = BAO and δ = OBC. The 3 internal angles of the ABC triangle are γ, γ + δ and δ. Since the sum of the angles of a triangle is equal to two right angles, we have

γ + ( γ + δ) + δ = 180°

then

2γ + 2δ = 180°

or simply

γ + δ = 90°

Q.E.D.

Converse

The converse of Thales' theorem is also valid; it states that a right triangle's hypotenuse is a diameter of its circumcircle.

The theorem and its converse can be expressed as follows:

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

Proof of the converse

This proof utilises two facts:

two lines form a right angle if and only if the dot product of their directional vectors is zero, and
the square of the length of a vector is given by the dot product of the vector with itself.

Let there be a right angle ABC and circle M with AC as a diameter. Let M's center lie on the origin, for easier calculation. Then we know

A = − C, because the circle centered at the origin has AC as diameter, and
(A − B) · (B − C) = 0, because ABC is a right angle.

It follows

0 = (A − B) · (B − C) = (A − B) · (B + A) = |A|² − |B|².

Hence:

|A| = |B|.

This means that A and B are equidistant from the origin, i.e. from the center of M. Since A lies on M, so does B, and the circle M is therefore the triangle's circumcircle.

The above calculations in fact establish that both directions of Thales' theorem are valid in any inner product space.

Generalization

Thales' theorem is a special case of the following theorem:

Given three points A, B and C on a circle with center O, the angle AOC is twice as large as the angle ABC.

The proof of this theorem is quite similar to the proof of Thales' theorem given above.

Application

Thales' theorem can be used to construct the tangent to a given circle that passes through a given point. (See figure.) Given a circle k with center a M, and a point P outside of the circle, we want to construct the (red) tangent(s) to k that pass through P. Suppose the (as yet unknown) tangent t touches the circle in the point T. From symmetry, it is clear that the radius MT is orthogonal to the tangent. So construct the midpoint H between M and P, and draw a circle centered at H through M and P. By Thales' theorem, the sought point T is the intersection of this circle with the given circle k, because that is the point on k that completes a right triangle MTP.

Since there the two circle intersect in two points, we can construct both tangents in this fashions.

History

Thales was not the first to discover this theorem since the Egyptians and Babylonians must have known of this empirically. However they did not 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 two right angles.

External link

Munching on Inscribed Angles

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