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In figure, seg EF is a diameter and seg ...

In figure, `seg` EF is a diameter and seg DF is a tangent segment. The radius of the circle is r. Prove that, `DExxGE=4r^(2)`.

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Line DF is tangent to the circle touching the circle at point F and line DGE is secant intersecting the circle at points G and E.
`:.` by tangent secant segments theorem,
`DF^(2)=DGxxDE " " "……..(1)"`
In `triangle DFE`,
`/_DFE=90^(@)" " "........"(" By tangent theorem ")`
`:.` Pythagoras theorem,
`DE^(2)=DF^(2)+EF^(2)`
`DE^(2)=DF^(2)+(2r)^(2)" " ".........."( :. " Diameter is twice the radius ")`
`:.DE^(2)=DF^(2)+4r^(2)`
`:.4r^(2)=DE^(2)-DF^(2)`
`:.4r^(2)=DE^(2)-DGxxDE" ""......."[" Form(1) "]`
`:.4r^(2)=DG(DE-DG)`
`:.4r^(2)=DExxGE" " '' ""......."(D-G-E )`
`:.DExxGE=4r^(2)`
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