Chapter 9.7, Problem 25WE

To determine

To prove:that $AX\cdot XB=CX\cdot XD$ with the help of given information.

Answer to Problem 25WE

  $AX\cdot XB=CX\cdot XD$ is proved

Explanation of Solution

Given information:

Two circles with a similar chord intersecting together $\overline{PQ}$

  $X$ be any point on $\overrightarrow{PQ}$

Through $X$ draw any chord $\overrightarrow{AB}$ of one circle.

Also draw through $X$ any chord $\overline{CD}$ of the other circle.

Calculation:

Consider the above information:

The figure we get,

  

Consider the theorem:

The product segment of one chord is equal to the product segment of the other chord when two chords meet within a circle.

  $\begin{array}{l}\ln \odot Y,PX\cdot XQ=AX\cdot XB\\ \ln \odot Z,PX\cdot XQ=CX\cdot XD\end{array}$

Now, solving the equations:

  $AX\cdot XB=CX\cdot XD$

Hence, $AX\cdot XB=CX\cdot XD$ is proved.