BodhiAI

Nov. 15, 2019

**Length of Common Chord**

**Let **

**and **

**be the two equations of the circles having centres and and radii r _{1 }and r_{2} respectively.**

**Let p _{1 }be the perpendicular distance from the centre C_{1} to that of the common chord.**

**Now, from the figure, we have**

** or **

** or **

Power of a Point

Given a point P and a circle, pass two lines through P that intersect the circle in points A, D and B,C respectively. Then AP.DP = BP.CP

The point P may lie either inside or outside the circle. The line through A and D (or that through B and C or both) may be tangent to the circle, in which case A and D coalesce into a single point. In all the cases, the theorem holds and is known as the Power of a Point Theorem.

When the point P is inside the circle, the theorem is also known as the Theorem of Intersecting Chords. When the point P is outside the circle, the theorem becomes the Theorem of Intersecting Secants.

The proof is exactly the same in all three cases mentioned above. Since triangles ABP and CDP are similar, the following equality holds

which is equivalent to the statement of the theorem: AP.DP = BP.CP.

The common value of the products is then depends only on P and the circle and is known as the Power of Point P with respect to the circle. Note that, when P lies outside the circle, its power equals the length of the square of the tangent from P to the circle.

i.e. PA.PB