3. TRIANGLE CEVIAN INVESTIGATIONS

These triangle investigations grew out of some problems posed by Steve Weissberg of Ithaca High School. He had students drawing various of the lines in a triangle, and asking questions about their intersection. After you get comfortable drawing altitudes, angle bisectors and so on, think about these.

1. Draw a triangle ABC. For which of the special lines angle bisector, median, altitude, can a triangle have two of the three the same, but not the third?

2. Altitudes line trace

Draw triangle ABC and altitudes from A and B. The intersection D is called the Orthocenter.

a. What do you think will happen to orthocenter D as you move point C around?

b. What is the locus of the orthocenter as C is constrained to move along a line parallel to base AB? (NOTE: you can attach C to a line you draw by selecting the line and C and using the edit menu.) Try animating the sketch and putting a TRACE on point D.

Construct the locus of D with respect to point C. Move C to see the different possibilities. What do you think this locus is? Can you prove it?

c. Same problem, but constrain C to move on a circle. Try various positions of the circle.

3. Triangle ABE has vertex E moving along a line DE parallel to base AB.

Angle bisector EF and median EM are drawn.

a. If E starts on the perpendicular bisector of AB and moves to the right, make a conjecture about what happens to point F.

What do you think happens as E moves to the right without end?

b. Draw the sketch on Sketchpad. What do you observe? Why?

c. Make a conjecture about the critical position(s) of E and prove it.

Discussion of angle bisector problem