Math 445 Wed 2/12 Class

Power Review from last time

Let O1 = (a, b) and O2 = (c, d).  If P = (x,y), write down the equations for:

• d1 = distance PO1

• d2 = distance PO2

If c1 is the circle of radius r1 and center O1, write the formula for:

• power of P with respect to c1.

If c2 is the circle of radius r2 and center O2, write the formula for:

• power of P with respect to c2.

If point P is a point on the radical axis, then the two powers are equal. 

• Write down the equation that says the two powers are equal and simplify. 

What kind of geometrical object is defined by this equation?  What can you now say about the radical axis?

Orthogonal Circles and Inversion

In this figure, the circles are orthogonal.  Let u be the circle with center A and v be the circle with center D. 

• If the radius of the circle with center A is R, tell the relationship among AB, AC and R.
• Solve this equation for AB.

Draw another secant through A intersecting circle v in points F and G. 

• Again, tell the relationship among AF, AG, and R and also solve the equation for AF.

Construct another circle w through B and C in the figure. 

• Is this new circle also orthogonal to the circle u? Explain why.

Definition of Inversion

You will be given the definition in class.  You may wish to write it here.

Construction problems (See Lab 6 and reading)

Given a circle c and a point B on c and a point A not on c. 

• Construct the circle through A and B orthogonal to c.

Given a circle c and a point A exterior to c. 

• Construct a point T on c so that AT is a tangent. 
• Then construct the circle with diameter AT.  Explain why this circle is orthogonal to c. 
• Also show where the inversion of A is in this figure.

From the stereographic projection figure shown in class:

• Construct the inversion of a point A in a circle c as stereographic projection of reflection in the equator.

Given two circles and a point A not on either circle,

• use inversion to construct the circle through A orthogonal to both circles.

Given 3 circles, show that the radical axes are concurrent or parallel. 

• Use this to construct a circle orthogonal to the given circles.