What Does Thales's Theorem State?

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Encyclopedia>Geometry>Planar Figures>Circles>Attributes>What Does Thales's Theorem State?

Thales’s theorem on inscribed angle

Rule

Thales’s Theorem

An inscribed angle that spans the diameter of a circle is always $90$ $^{\circ}$ $(u + v = 90^{\circ})$ .

This makes Thales’s theorem a special case of the inscribed angle theorem. If the inscribed angle is

90

^{\circ}

, then the central angle is

180

^{\circ}

. This is correct because the diameter can be thought of as an angle of

180

^{\circ}

Think About This

Proof of Thales’s Theorem

From the figure, you can see that the triangles

△ B A P

and

△ C A P

are both isosceles triangles, because two of the sides of both triangles are the radius of the circle. You can therefore write the sum of the angles in the red triangle as follows:

\begin{array}{l} u + v + v + u & = 180^{\circ} \\ 2 u + 2 v & = 180^{\circ} \\ u + v & = 90^{\circ} \end{array}

Q.E.D

Example 1

Find all the angles of the triangle, where $A B$ is the diameter

Since $A B$ is the diameter, Thales’s theorem tells us that $∠ A C B = 90^{\circ}$ . You also know that $S B = S C = S A$ is the radius of the circle. Hence $△ B S C$ is isosceles and $∠ S C B = 35^{\circ}$ . You therefore find that

∠ B S C = 180^{\circ} - 35^{\circ} - 35^{\circ} = 110^{\circ}

You know that $△ A S C$ is isosceles and that $∠ A S C$ is the supplementary angle of $∠ B S C$ . Hence

∠ A S C = 180^{\circ} - 110^{\circ} = 70^{\circ}

Furthermore,

∠ S A C = ∠ S C A = \frac{180^{\circ} - 70^{\circ}}{2} = 55^{\circ}

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Circle of Apollonius