Assignment 2B (Due Monday, 10/13)

Note: "Construct" means construct with compass and unmarked straightedge or else with Sketchpad. (If you do a Sketchpad construction, it should be your own, not a team effort from lab.)

Note: There are some changes from the "printed packet version." A note was added to 2.4 and 2.6 was reworded for clarity.

2.4 Some special lines of a triangle

Look up the definitions of some special lines of a triangle: the perpendicular bisectors of the sides, the altitudes, the interior and exterior bisectors of the angles, and the medians.

• For the two triangles ABC in this Special Lines in a Triangle Worksheet, carry out the constructions of some special lines -- the perpendicular bisector of BC, the angle bisector of A, the altitude through A, and the median through A.
• Then answer the questions on the worksheet. (Note: It is OK to draw your own triangles instead of printing these out, but the should be about the same shape and size as the ones on the worksheet -- and the second one should be isosceles.)

2.5 Incircles and Excircles of a triangle

Draw a "general" triangle ABC (i.e., one that is not "special", not isosceles, equilateral, right, or having some special angle).

• Construct carefully the interior and exterior angle bisectors of the triangle.
• Construct carefully with straightedge and compass (or with Sketchpad -- do your own independent drawing) the incircle and the 3 excircles of the triangle.
• Explain why the interior and exterior bisectors of an angle are perpendicular.
• If P, Q, R are the centers of the excircles, what special lines of the triangle PQR are the interior angle bisectors of ABC?

2.6 Locus theorem for equal distances to intersecting lines

Let m and n be two lines that intersect at O. Prove that the set of points P that are at equal (positive) distance from m and n is a set consisting of the two lines that bisect the four angles m and n form at O.

2.7 Composition of 2 line reflections

Suppose that p and q are two lines that intersect at O at an angle of 60 degrees. Let A be any point distinct from O. Let A' be the reflection of A in p and let A'' be the reflection of A' in q.

• Construct a figure that shows p and q and a good example of A and its images.
• What is the measure of angle AOA''? Prove that this is true.
• If A''' is the reflection of A'' in p, explain why A''' is in fact always the reflection of A in a certain line (independent of the choice of A). Construct this line and explain why this is true.
• In a separate figure, construct an interesting example of a kaleidoscopic figure formed by two lines meeting at 60 degrees.