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.

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

Verify Let’s verify that this always works. Drag point B around the circle. Does the measurement stay at 90°?

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Application We can use Thales’ Theorem to construct the tangent to a circle that passes through a given point 2. 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, (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.

Construction So 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