Study Questions from Week 3
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
You should be able to prove each of these theorems and
- 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
- 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
- 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
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
- 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.
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.
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