Thales' theorem

The theorem of Thales is a set of geometry and a special case of the circle angle set. The first proof is attributed to the ancient Greek mathematician and philosopher Thales of Miletus. The statement of the theorem was previously known in Egypt and Babylonia.

Formulation of the theorem and its converse

Brief Formulation: All angles on semicircle are right angles.

Exact wording: Constructed to a triangle of the two endpoints of the diameter of a semicircle ( Thales circle) and another point of this semi-circle, so you always get a right triangle.

Or: If the point C of a triangle ABC on a semicircle above the segment AB, then the triangle always has a right angle at C.

The reversal of the set is correct: the center of the circumference of a right triangle is always the center of the hypotenuse, which is the longest side of the triangle, which is right angle to.

Or: If the triangle ABC with C at a right angle, so C lies on a circle with the hypotenuse of AB as a diameter.

Evidence

Euclid derives the theorem of Thales in the third volume of its elements using the following sentences, which are also attributed to Thales and are included in the first volume, here:

• In any isosceles triangle the angles at the base are equal.
• The sum of the angles in a triangle is 180 °.

ABC is a triangle within a circle of [AB ] as a circle diameter and the radius r. Then the midpoint M of the segment [ AB ] is also the center of the circle. The track lengths [AM ], [ BM ] and [ CM] are therefore equal to the radius r.

The route [ CM] divides the triangle ABC into two triangles AMC and BCM, which are isosceles. The base angles of these triangles, so the angles at the base side [AC ] and [ BC], are thus respectively equal to (or in the figure).

The sum of the angles in the triangle ABC is 180 °:

Dividing this equation on both sides by 2, we obtain

Thus we have shown that the angle with vertex C is a right angle.

The converse of the theorem of Thales can be traced back to the statement that the diagonals of a rectangle are equal in length and bisect each other.

Applications

An important application of the Thales circle is the construction of the two tangents to a circle k by a lying outside this circle point P:

Since the upper passing through P tangent to the circle k touches exactly at the point T, the triangle OPT has a right angle at the point have T, or in other words: The distance [ OT ] must be perpendicular to the line TP.

The two points O and P are given. From the point T, we only know that it must be somewhere on the circle k. If one were to consider only this condition, you could draw an infinite number of triangles OPT.

To find a triangle OPT, who is also rectangular, we determine by intersection with the perpendicular bisectors of the center H of the segment [ OP ], draw a semi-circle ( with center H) over the range [OP ] and make us the principle of the Thales circle advantage All triangles with the base side [OP ], whose third vertex lies on the circle of Thales, are rectangular. This is also true for the triangle OPT.

The contact point T can therefore lie on the intersection of the circle k with the light gray circle of Thales. By connecting P and T is now obtained the required tangent (in the drawing red).

There is a second, symmetrical solution in the lower half of the circle. The tangent T'p ( also shown in red ) touches the circle also, in the point T '.

A further application is the quadrature of the rectangle.

Construction of real square roots

Using the theorem of Thales can be the square root of construct, since it applies. Substituting, and constructs a Thales circle, so is the height of the right triangle equals the square root of.