Facts to prove from Exploration 3.1

Proposition 1.

Let b be a circle with center B and c with a circle with center C.  Suppose the circles intersect at points A and A’, with A not on line BC.  Then:

• Prove that triangle ABC is congruent to triangle A’BC.
• Prove that line BC is the perpendicular bisector of AA’.

Proposition 2.

Given a segment AB and a circle c with center C that passes through A and B, prove that C is on the perpendicular bisector of AB.

Proposition 3.

Given a segment AB and perpendicular bisector m of AB.  Prove that if C is any point on m, then the circle with center C through A must also pass through B.

Facts to prove from Exploration 3.2

Given points A and B and two circles of the same radius r – circle a has center A and circle b has center B.  Any point P of intersection of a and b is a point P so that the

Distance PA = distance PB.  Prove that P is on the perpendicular bisector of AB.  Tell why this explains the observation from Investigation 1 that the trace of the intersection points of the ripples is a line.

In the 3 Ripples Investigation, explain why – IF there is a point Q where the 3 loci are concurrent, that the point Q is equidistant from the vertices of the triangle.

Facts to prove from Exploration 3.3

Prove that for a triangle ABC, if P is the point of intersection of the perpendicular bisectors of AB and BC, that P is also on the perpendicular bisector of CA.

Explain how to construct a circle through A, B, and C and then tell why there is only one circle and why the construction works.