Examine each of the following 13 explorations. Select anythreeof them for write-ups.Due October 3, 2000.

0. In class we investigated the relationship between a triangle and its medial triangle areas. We proved our

conjecture in class. Extend this idea and instead of medial triangles why don't you trisect the sides?

quarter the sides? nth the sides? See if you can find a pattern. See if you can prove any of the conjectures you make.

And/or...take the idea of medial triangle and extend it to quadrilaterals, pentagons...n-gons.

1. Construct a triangle and its medians. Construct a second triangle with the three sides having the lengths of the three medians from your first triangle. Find some relationship between the two triangles.

(E.g., are they congruent? similar? have same area? same perimeter? ratio of areas? ratio or perimeters?)

Extension: What if you trisect the sides? quarter the sides? nth the sides?

2. If the original triangle is equilateral, then the triangle of medians is equilateral. Will an isoceles original triangle generate and isoceles triangle of medians? Will a right triangle always generate a right triangle of medians? What if the medians triangle is a right triangle? Under what conditions will the original triangle and the medians triangle both be right triangles?

3. Given line segments j, k, m. If these are the medians of a triangle, construct the triangle. Explain

your construction.

4. Given three points A, B, and C. Draw a line intersecting AC in the point X and BC in the point Y such that

AX = XY = YB

5. Construct the common tangents to two given circles. Make a script. Test it for all the different cases.

6. Consider any triangle ABC. Find a construction for a point P such that the sum of the distances from P to each of the three vertices is a minimum. What about any quadrilateral ABCD? See the Water Supply Problems in your text!

7. You have investigated, found, and hopefully proved constructions of the balancing point of triangle and of a quadrilateral. What about for a pentagon? a hexagon? can you generalize for any n-gon?

8. Investigate the Light Ray in a Triangle Problem (Pgs 90 - 94) in your text.

9. Construct any triangle ABC and its orthocenter H. Construct D, E, and F as the intersections of the opposite sides and the "diagonals" of Quadrilateral ABCH. Where are the points D, E, and F? Are the same three points located if the labels for points A and H are interchanged? Why?

10. Examine the triangle formed by the points where the extended altitudes meet the circumcircle. How is it related to the Orthic triangle?

11. Construct any acute triangle ABC and its circumcircle. Construct the three altitudes ha, hb, and hc. Extend each altitude to its intersection with the circumcircle and let Ha, Hb, and Hc be the corresponding segments from the vertex to the intersection with the circumcircle. Find

and prove your result.

12. Find the triangle of minimal perimeter that can be inscribed in a given triangle. (For a start, you may want to restrict your investigation to the given triangle being acute.)