Thales’ Theorem

Easily Constructible Right Triangle

Draw a circle.

Draw a line using the circle’s center and radius control points.

Construct the intersection of the line and circle.

Label the intersection points A and C.

ConstructionSo Far

Finishing the Construction

Draw a third point somewhere on the circle. Label this point B.

Connect the three points on the circle with line segments to form a triangle.

Measure ∠ABC.

Result

Thales’ Theorem

Thale’s Theorem: An inscribed angle in a semicircle is a right angle1.

1 Weisstein, Eric W. “Thales’ Theorem.” From Mathworld--A Wolfram Web Resource.

http://mathworld.wolfram.com/ThalesTheorem.html

Verify

Let’s verify that this always works.

Drag point B around the circle.

Does the measurement stay at 90°?

Create a New Document

Application

We can use Thales’ Theorem to construct the tangent to a circle that passes through a given point2.

Start by drawing a circle and a point outside of the circle.

Label the circle’s center O and the point P.

2 Wikipedia contributors, ‘Thales’ theorem’, Wikipedia, The Free Encyclopedia,

http://en.wikipedia.org/w/index.php?title=Thales%27_theorem&oldid=417586850

(accessed March 18, 2011).

Initial Figure

Application (Cont.)

Draw the line segment OP.

Construct the midpoint of OP and label it H.

Draw a circle with center H and radius P.

ConstructionSo Far

Application (Cont.)

Construct the intersections of the circles.

Label the intersections S and T.

Draw the lines PS and PT.

Note how these lines are tangent to the original circle!

Result

Application (Cont.)

Measure ∠OSP and ∠OTP.

Can you see the use of Thales’ Theorem?

Where else might Thales’ Theorem be useful?

Conclusion