Thales’ Theorem and Thales Circle
Thales’ theorem states that if A and B are the ends of a diameter of a circle and C is any other point on the circle, then the angle ACB is a right angle; the circle with the segment AB as diameter is called the Thales circle. In brief: every angle inscribed in a semicircle is 90°. The converse is also true: the locus of points C from which a segment AB is seen at a right angle is the circle with diameter AB (without the points A and B). For AB = 10 the circle has radius 5.
The theorem gives a simple way to construct right angles, to find tangents from an external point and to decide whether a triangle is right-angled. Note that the name “Thales’ theorem” is also used in some countries for the intercept theorem about parallel lines and proportional segments, so check which meaning your textbook uses. This guide explains the theorem about the semicircle, gives a proof, shows the converse and applications and lists the main facts in a table.
What does Thales’ theorem say?
Let AB be a diameter of a circle with center M, and let C be a point of the circle different from A and B. Then ∠ACB = 90°. In other words, a triangle inscribed in a circle with one side as the diameter is a right triangle, and the diameter is its hypotenuse. The circle is the circumcircle of the right triangle, and its center is the midpoint of the hypotenuse.
How do you prove it?
Connect C with the center M. Since MA = MB = MC (radii), triangles AMC and BMC are isosceles. Let ∠MAC = α. Then ∠MCA = α, since the base angles of an isosceles triangle are equal. Let ∠MBC = β. Then ∠MCB = β. The angle sum in triangle ABC gives α + β + (α + β) = 180°, so 2(α + β) = 180°, and α + β = 90°. But ∠ACB = ∠MCA + ∠MCB = α + β = 90°. ∎
Coordinate check
Take the circle x2 + y2 = 25 with A(−5, 0) and B(5, 0). The point C(3, 4) lies on it. The vectors CA = (−8, −4) and CB = (2, −4) have dot product −16 + 16 = 0, so they are perpendicular. ✓ And the slopes: CA has slope (−4)/(−8) = 1/2 and CB has slope (−4)/2 = −2, with product −1.
What is the converse?
If angle ACB is a right angle, then C lies on the circle with diameter AB. Proof: let M be the midpoint of the hypotenuse AB. In a right triangle the median to the hypotenuse equals half the hypotenuse, so MC = MA = MB, and C lies on the circle with center M and radius AB/2.
Therefore the Thales circle is the locus of all points from which AB is seen at 90°. Points inside the circle see AB at an obtuse angle, and points outside see it at an acute angle.
How do you use the theorem?
Example 1: finding a missing angle
AB is a diameter and C is on the circle. If ∠CAB = 35°, then ∠ABC = 90° − 35° = 55°, and ∠ACB = 90°.
Example 2: finding a length
A circle has diameter AB = 26. Point C is on the circle with AC = 10. Then BC = √(262 − 102) = √576 = 24, by the Pythagorean theorem in the right triangle ACB.
Example 3: constructing a tangent from an external point
Given a circle with center O and an external point P. Construct the midpoint M of OP and draw the circle with center M through O and P. It meets the given circle at the points T1 and T2. Since OP is a diameter of the new circle, ∠OTP = 90° at each point of intersection, so PT is perpendicular to the radius OT and is therefore a tangent. For OP = 13 and radius 5, PT = √(169 − 25) = 12.
Example 4: constructing a right triangle
To construct a right triangle with hypotenuse 10 and one leg 6: draw AB = 10, draw the circle with diameter AB, and the circle with center A and radius 6. Their intersection C gives AC = 6, ∠C = 90° and BC = 8.
Example 5: testing with coordinates
Is the triangle with vertices A(1, 1), B(7, 1) and C(4, 4) right-angled at C? The midpoint of AB is M(4, 1) and AB/2 = 3. The distance MC = 3, so C lies on the circle with diameter AB. By the converse, ∠C = 90°. Check with slopes: CA has slope (1 − 4)/(1 − 4) = 1 and CB has slope (1 − 4)/(7 − 4) = −1, and the product is −1 ✓.
Facts about the Thales circle
| Position of C | Angle ACB | Distance from the midpoint M compared with AB/2 |
|---|---|---|
| On the circle | 90° (right) | equal |
| Inside the circle | more than 90° (obtuse) | smaller |
| Outside the circle | less than 90° (acute) | larger |
| At A or B | not defined | equal |
Where does the theorem come from?
The theorem is named after Thales of Miletus, a Greek philosopher who lived in the sixth century BCE. Ancient sources, such as Proclus, attribute to him several geometric results, including this one, but there is no surviving text from Thales himself, and the claim is uncertain. Some historians think the result was already known in Babylonian mathematics, but this is debated. A rigorous proof appears in Euclid’s Elements (Book III, Proposition 31).
Where is it used today?
Carpenters check a right angle by placing a corner on a semicircle. A set square can be tested by putting its right-angle vertex on a circle whose diameter is the hypotenuse. In engineering drawing, tangents from a point are constructed with the Thales circle. The idea is used in geometry software to build right angles, and in computer graphics to test whether a point lies inside a circle that has a given segment as its diameter.
Which mistakes are common?
Applying the theorem to a chord that is not a diameter
The angle in a semicircle is a right angle only if the base of the angle is a diameter. For other chords, the inscribed angle changes with the chord.
Forgetting the endpoints
The points A and B themselves do not form a triangle. The Thales circle excludes them.
Mixing up the two Thales theorems
The intercept theorem is about parallel lines. The semicircle theorem is about the right angle in a circle.
Using the diagram instead of the diameter condition
A figure may look like a semicircle. The theorem needs AB to pass through the center.