Given: seg EF is the diameter. seg DF is a tangent to the circle, radius = r
To prove: DE × GE = 4r Construction: Join seg GF.
Proof:
seg EF is the diameter. [Given]
∴ ∠EGF = 90° (i) [Angle inscribed in a semicircle] seg DF is a tangent to the circle at F. [Given]
∴ ∠EFD = 90° (ii) [Tangent theorem] In ∆DFE,
∠EFD = 90 ° [From (ii)]
seg FG ⊥ side DE [From (i)]
∴ ∆EFD ~ ∆EGF [Similarity of right angled triangles]
∴ (EF/GE) = (DE/EF) [Corresponding sides of similar triangles]
∴ DE × GE = EF2
∴ DE × GE = (2r)2 [diameter = 2r]
∴ DE × GE = 4r2