Comments on class presentations, September 8, 10, 12

Laura's group:

In the construction of a circle by center and radius, the circle W is not needed. (Drawing it confused several students.)

The proof of the construction of an arc (circle) through three points is incorrect. In the proof presented, the existence of the circle is assumed! To prove that the point O is the center of a circle passing through the points A, B, and C, you have to show that the distances OA, OB, and OC are equal. This is done by proving that the triangles OAB and OBC are isosceles.

Becky's group:

The construction of the regular pentagon presented in class was confusing to many students. (Charnelle contributed a simpler proof.) In class 9/15 I presented the simplest construction I know of the regular pentagon. It depends on the theorem that the ratio of a diagonal to a side of a regular pentagon is the golden ratio.

Laura S's group:

Inclusive definitions of types of quadrilaterals imply that every parallelogram is a trapezoid, for a trapezoid is a quadrilateral with at least one pair of opposite sides. (Most books say that a trapezoid has exactly one pair of parallel sides.)

Michelle's group:

It would be simpler just to use the facts proved by Laura S's group, rather than to reprove them.

Eric's group:

Note that the proof of the concurrence of the perpendicular bisectors is related to the proof of the construction of the circle through three points.

Allyson's group:

I don't know a proof of the concurrence of the altitudes that doesn't use a trick (a surprising construction).