Chapter 14, Problem 7PE

To determine

To prove: if two coplanar circles intersect in two pints, then the line joining those points bisects a common tangent segment.

Explanation of Solution

Given information:

Two circles center at $A,B$ intersect at two points $I,J$ have a common tangent $EF$ .

  

Calculation:

Consider the mid-point of $AB$ as $R$ . Since $\overline{EF}=\overline{AB}$ , we have to show K is lies on EF. Let $DC$ be the intersecting line.

For circle of center $A$ ,

  $\overline{AR}\perp \overline{DC}$ , then $\overline{AR}=\overline{EK}$ and $\overline{AR}\left|\right|\overline{EK}$ implies $\overline{EK}\perp \overline{DC}\mathrm{....................}(i)$

Similarly for circle of center $B$ ,

  $\overline{BR}\perp \overline{DC}$ , then $\overline{KF}=\overline{RB}$ and $\overline{KF}\left|\right|\overline{RB}$ implies $\overline{KF}\perp \overline{DC}\mathrm{........................}(ii)$

Equation (i) and (ii) indicating $R$ lies on the segment $AB$ , and together $K$ is lies on the tangent line $EF$ .

Hence proved the requirement.