PCMI 5 Minute Short

From GeoGebraWiki

This all started last year with a rather simple angle chasing problems from the Mathematics 2 curriculum at Phillips Exeter Academy, 2007, p. 32)):

Triangle ABC has a 34-degree angle at A. The bisectors of angles B and C meet at point I. What is the measure of angle BIC? Answer this question assuming that triangle ABC is (a) right; (b) isosceles; (c) scalene.

Explore the 34-degree case

Since then, I have worked with Todd Edwards at Miami (OH) University and Bob Klein at Ohio University in writing up this problem in hopes of getting it published (SOMEWHERE), the purpose being to explore one possible role of dynamic geometry in support of teacher questioning. In particular, we are applying the What-If-Not (WIN) questioning strategy advocated by Brown and Walter (2004) promote the following investigations.

Brown, S. I., & Walter, M. I. (2004). The art of problem posing (3rd Ed.). Hillsdale, NJ: L. Erlbaum Associates.

Contents 1 Incenter 2 Circumcenter 3 Orthocenter 4 Centroid 5 Nine-Point Center 6 Symmedian Point 7 Gergonne Point 8 Some Clever Restrictions 9 Park City Mathematics Institute

Incenter

Generalized Triangle Incenter Task: Triangle ABC has an angle of measure x degrees at A. Let I be the incenter of triangle ABC. Describe the measure of angle BIC in terms of x. Incenter

Circumcenter

Generalized Triangle Circumcenter Task: Suppose triangle ABC has an x-degree angle at A. Let O be the circumcenter of ABC. Describe the measure of angle BOC in terms of x. Circumcenter

Orthocenter

Generalized Triangle Orthocenter Task: Suppose triangle ABC has an angle of measure x degrees at A. Let H be the orthocenter of triangle ABC. Describe the measure of angle BHC in terms of x. Orthocenter

Centroid

Suppose triangle ABC has an angle of measure x degrees at A. Let G be the centroid of triangle ABC. Describe the measure of angle BGC in terms of x. Centroid

Nine-Point Center

Suppose triangle ABC has an angle of measure x degrees at A. Let N be the nine-point center of triangle ABC. Describe the measure of angle BNC in terms of x.Nine-Point Center

Symmedian Point

Gergonne Point

Gergonne Point

Some Clever Restrictions

• Triangle Vertices on Rectangular Hyperbola
• Centroid: Vertex A on Perpendicular Bisector of BC
• Constant Centroid Angle

Park City Mathematics Institute

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