## by Martin Flashman

I. Tangent and Extreme Problems

1. The Tangent Problem in Euclid- Tangent to a circle.
1. Euclid's Proof
2. Application of the proof to construction of a tangent. (Use GSP.)
3. What this means with coordinates.

2. The Tangent Problem in Archimedes- Tangent to a parabola
1. Archimedes statements - Missing proofs. (Use GSP.)
2. Application of the statement to construction of a tangent. (Use GSP.)
3. What this means with coordinates.
4. A coordinate based proof and a maximization/minimization problem.

3. A Minimization Problem. Old MacDonald's Farm (Use GSP.)
1. Exploration with Geometers' Sketchpad.
2. Locus of areas yields dynamic view of solution.
3. Changing the constraint leads to conjecture on the general solution.
4. Variables lead to a parabola and the problem solution (sans calculus).
5. Generalizations of this problem with GSP.
Triangle (1 point), rectangle (2 point) , ... ? (Use GSP.)

4. Closest point on a curve to a given point.
1. Euclid: Line.
2. Euclid: Circle.
3. Archimedes: Parabola.
4. Exploration with GSP. (Use GSP.)

5. The 2 Locations 1 Utility on Line Connection Problem.
1. Exploration with GSP. (Use GSP.)
2. Locus of lengths yields view of solution. (Use GSP.)
3. The reflection solution. (Use GSP.)
4. Some generalizations:
1. 3 locations, 4 locations,.... (Use GSP.)
2. Connection on a curve. (Use GSP.)

6. The Runner/Swimmer Problem
1. Exploration with GSP. (Use GSP.)
2. Locus of time yields view of solution.
3. Comparing Solutions
4. A generalization: Triatholon Problem?

7. The Least Squares Problem
1. Exploration with JavaSP.
2. Locus of sums yields view of solution.
4. Some generalizations. - Other curves.

II. Area and Velocity Problems

1. Area of a Parallelogram and Triangle in Euclid. Web site. (Math 371)
2. Area of a Circle in Euclid. Web site. (Math 401)
3. Area of a parabola in Archimedes. (Math 401)
4. Position of an object with constant acceleration
- Oresme and Galileo (Math 401)

5. Transformation Figures (Current - Work in Progress)
1. Visualization of uniform motion and position. (Use GSP.)
2. Constant velocity. (Use GSP.)
3. Visualization of uniform acceleration. (Falling objects?)

Euclid (Thanks to D. Joyce - Clark U.)
Book III (Circles)

The straight line drawn at right angles to the diameter of a circle from its end will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed, further the angle of the semicircle is greater, and the remaining angle less, than any acute rectilinear angle.

Corollary to 16. From this it is manifest that the straight line drawn at right angles to the diameter of a circle from its end touches the circle.

From a given point to draw a straight line touching a given circle.

If a straight line touches a circle, and a straight line is joined from the center to the point of contact, the straight line so joined will be perpendicular to the tangent.

If a straight line touches a circle, and from the point of contact a straight line is drawn at right angles to the tangent, the center of the circle will be on the straight line so drawn.