Study Questions from Week 3

Concurrence Theorems

Given a triangle ABC, there are a number of special lines that occur in threes, and it happens that they may always be concurrent. Here are the four instances of this that we have encountered in the course. Each one has its own character and has a different proof.

You should be able to prove each of these theorems and associated problems.

Perpendicular Bisectors

This was done in Week 2. The proof involves distances to points. The point of concurrence is called the circumcenter. This leads to the construction of the circumcircle. The special case of the right triangle is related to the reason that the Carpenter Construction is true.

Medians

The proof was done in class; it involves midpoint figures and parallelogram properties. Major hints are also in B&B and Bix readings. The point of concurrence is called the centroid. (This is the center of gravity of ABC, a fact which we have not discussed yet.)

Altitudes

This is proved from the Perpendicular Bisector case by constructing a new triangle whose midpoint triangle is the original one. The main part of the proof then is the construction of this new, bigger triangle and checking that the original really is the midpoint triangle. The point of concurrence of the altitudes is called the orthocenter.

Angle Bisectors.

The proof involves distances to lines. There are really four points of concurrence, one (called the incenter) for the three bisectors of interior angles and the other three (called excenters) for a pair of exterior angle bisectors and one interior.

Line Distance, Angle Bisectors, Strips and Incircles

Chapter 5 of GTC lays out these ideas. You should be able to discuss or prove. Also look in B&B and Bix.

• Given a circle with center E tangent to two lines (intersecting or parallel). What can you say about the distance from E to the lines.
• Given two lines intersecting at A and a circle with center E tangent to both lines, how is ray AE related to the two lines.
• Given a line and a distance d, what is the set of points at distance d from the line?
• Given a distance d and two lines intersecting at A, describe the set of points at distance d from both of the lines. What kind of figure do you get?
• Given a triangle ABC, prove that there exists a point J (the incenter) which is equidistant from the three sides (the three segments AB, BC, and CA). Also prove that there are 4 points (the incenter and 3 excenters) equidistant from the three sides extended.
• Explain how these four points can be used to construct circles tangent to the sides (or sides extended) of triangle ABC.
• Given any three lines, what are the possible ways that they can meet (form a triangle extended, are all parallel, other options)? How many circles are there tangent to all three lines in each case. How are the circles constructed?
• Given a triangle ABC and its 3 excenters P, Q, R, explain how triangle PQR has its orthocenter at J, the incenter of ABC, and the feet of its altitudes at A, B, and C.

Line Reflection

You should be able to find the lines of symmetry of simple figures. Also define what is meant by line symmetry and explain why a triangle is reflected to a congruent triangle.