Question

In the given figure two circles intersect each other in points A and B.Two tangents touch these circles in points P,Q and R,S as shown.line AB intersects seg PQ in C and seg RS in D.Show that C and D are the mid points of seg PQ and seg RS respectively.

Answer 1

$$\begin{gathered} \mathrm{CP}\mathrm{is}\mathrm{a}\mathrm{tangent}\mathrm{and}\mathrm{CD}\mathrm{is}\mathrm{a}\mathrm{secant}. \\ By\tan gentsecanttheorem,\mathrm{we}\mathrm{have}: \\ {\mathrm{CP}}^2=\mathrm{CA}\times \mathrm{CD}...\left(1\right) \\ \mathrm{CQ}\mathrm{is}\mathrm{a}\mathrm{tangent}\mathrm{and}\mathrm{CD}\mathrm{is}\mathrm{a}\mathrm{secant}. \\ \mathrm{By}\mathrm{tangent}\mathrm{secant}\mathrm{theorem},\mathrm{we}\mathrm{have}: \\ {\mathrm{CQ}}^2=\mathrm{CA}\times \mathrm{CD}...\left(2\right) \\ \mathrm{Using}\mathrm{eq}s.\left(1\right)\mathrm{and}\left(2\right),\mathrm{we}\mathrm{have}: \\ {\mathrm{CP}}^2={\mathrm{CQ}}^2 \\ \Rightarrow \mathrm{CP}=\mathrm{CQ} \\ \mathrm{Hence},\mathrm{C}\mathrm{is}\mathrm{the}\mathrm{midpoint}\mathrm{of}\mathrm{PQ}. \end{gathered}$$

$$\begin{gathered} \mathrm{RD}\mathrm{is}\mathrm{a}\mathrm{tangent}\mathrm{and}\mathrm{DC}\mathrm{is}\mathrm{a}\mathrm{secant}. \\ \mathrm{By}\mathrm{tangent}\mathrm{secant}\mathrm{theorem},\mathrm{we}\mathrm{have}: \\ {\mathrm{RD}}^2=\mathrm{DB}\times \mathrm{DC}...\left(3\right) \\ \mathrm{CQ}\mathrm{is}\mathrm{a}\mathrm{tangent}\mathrm{and}\mathrm{CD}\mathrm{is}\mathrm{a}\mathrm{secant}. \\ \mathrm{By}\mathrm{tangent}\mathrm{secant}\mathrm{theorem},\mathrm{we}\mathrm{have}: \\ {\mathrm{DS}}^2=\mathrm{BD}\times \mathrm{DC}...\left(4\right) \\ \mathrm{Using}\mathrm{eqs}.\left(3\right)\mathrm{and}\left(4\right),\mathrm{we}\mathrm{have}: \\ {\mathrm{RD}}^2={\mathrm{DS}}^2 \\ \Rightarrow \mathrm{RD}=\mathrm{DS} \\ \mathrm{Hence},\mathrm{D}\mathrm{is}\mathrm{the}\mathrm{midpoint}\mathrm{of}\mathrm{RS}. \end{gathered}$$