# Finding the center of a circle

• Dec 2nd 2009, 08:37 PM
gunjannigam
Finding the center of a circle
Given
1)A point A(x,y).
2)Two others points P(x1,y1) and Q(x2,y2)

Find
The equation(basically the center) of circle that is tangent to line AP and AQ and of given radius r.

How to solve it.
• Dec 2nd 2009, 10:03 PM
earboth
Quote:

Originally Posted by gunjannigam
Given
1)A point A(x,y).
2)Two others points P(x1,y1) and Q(x2,y2)

Find
The equation(basically the center) of circle that is tangent to line AP and AQ and of given radius r.

How to solve it.

1. The 3 points should not be placed on one single straight line.

2. Let $\displaystyle M\left(x_M,\ y_M\right)$ be the midpoint of the circle. The distance of M to AP or AQ must be |r|.

3. Consequently you get 4 pairs of equations for the 4 different Ms:

$\displaystyle d(M, AP) = r~\wedge~d(M, AQ) = r$

$\displaystyle d(M, AP) = r~\wedge~d(M, AQ) = -r$

$\displaystyle d(M, AP) = -r~\wedge~d(M, AQ) = r$

$\displaystyle d(M, AP) = -r~\wedge~d(M, AQ) = -r$

4. Solve each pair of equations for $\displaystyle \left(x_M,\ y_M\right)$