1. Thales’ Theorem

2. 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.

3. Construction So Far

4. 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.

5. Result

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

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

8. Create a New Document

9. 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, http://en.wikipedia.org/w/index.php?title=Thales%27_theorem&oldid=417586 850 (accessed March 18, 2011). http://en.wikipedia.org/w/index.php?title=Thales%27_theorem&oldid=417586 850

10. Initial Figure

11. 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.

12. Construction So Far

13. 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!

14. Result

15. Application (Cont.) Measure ∠ OSP and ∠ OTP. Can you see the use of Thales’ Theorem? Where else might Thales’ Theorem be useful?

16. Conclusion