Challenge Problems

Various links to Challenge problems are given here. Since many of these are links to other web pages, you may need to use the Back command on your browser to return to this page.

1. Treasure Island Problem.

2. Nine Point Circle. This is an essay with some proofs omitted -- for you to provide. Prove: For any triangle there exists a Nine Point Circle -- a circle that passes through the three midpoints of the sides, the three feet of the perpendiculars to the sides, and the three midpoints of the segments from the orthocenter to the vertices.

3. Morley's Theorem. Link to an interesting web site on "Morley's Miracle." Prove: The three points of intersection of the adjacent trisectors of the angles of any triangle form an equilateral triangle.

4. Path Problems

Minimum Paths -- e.g. the Hiker's Path, the billiards problem, etc.

Angle of Incidence and Angle of Reflection

5. Shortest Highway

6. Steiner's Minimum Distance Problem

7. Pythagorean Theorem

8. Theorem of Pappas

9. Simson Line

10. Theorem of Menalaus

11. 20-30-150 Triangle Problem

12. Bisectors of a 120 degree obtuse triangle

13. Triangle and its medians

14. 100 Degree Isosceles Triangle

15. Number of Pieces SPACE can be cut into with 5 planes?