§6.1 Constructions The student will learn about: more Euclidean constructions. 1

Proofs of Constructions For homework you proved that the angle bisector construction works. You should be able to prove that any of the previous constructions work. 2

Paper Folding Another way to do these constructions without Euclidean tools is with paper folding. You should be aware that there is an art of paper folding called origami. Tom Hall, a professor at Western New England College has done a lot in the area of mathematical research in origami. 3

Bisect a Segment Paper folding. 4

Bisect an Angle 5 Paper folding.

Perpendicular to a Point on the Line 6 Paper folding.

Perpendicular to a Line from a Point not on the Line 7 Paper folding.

Parallel to a Line from a Point not on the Line 8 Paper folding.

Sum of the Angles of a Triangle 9 Paper folding.

Area of a Triangle 10 Paper folding.

Incenter/Circumcenter 11 Paper folding.

Orthocenter/Centroid 12 Paper folding.

Homework 13 Do a compass and straight edge construction of the incenter, circumcenter, orthocenter and centroid.

Nine Point Circle. 14 How many points are needed to uniquely define a circle? How unusual might it be to have say five special points on the same circle? Would you believe that there is a special circle defined by nine points associated with the same triangle?

Nine Point Circle. 15 The three medians. The three altitude feet. The three points on the altitudes that are midway between the vertex and the orthocenter. Pretty cool!

Homework 16 Do a compass and straight edge construction of the nine point circle finding its center.

Assignment: §6.1