As shown in the figure, BD is the diameter of circle O, e is a point on circle O, the straight line AE intersects the extension line of BD at point a, BC ⊥ AE at point C, and ∠ CBE = ∠ DBE. (1) try to explain that AC is the tangent line of circle O. (2) if the radius of circle O is 2, AE = 4, root 2, find the length of de

When OE is connected, there is ∠ OEB = ∠ OBE. It is known that ∠ CBE = ∠ DBE, so ∠ OEB = ∠ CBE, and OE ‖ BC, ∠ OEA = 90 °
∵∠OEA=90°
∴OE⊥AC
If e is a point on O, then AC is tangent
2）
∵OE⊥AC
∴AO=√（AE²+OE²）=√[（4√2)²+2²]=9,AB=AO+OB=9+2=11.
BC = OE · AB / AO = 2 × 11 / 9 = 22 / 9
It is known that ∠ CBE = ∠ DBE, so RT △ BCE ∽ RT △ bed, then BC / be = be / BD, be & sup2; = BD · BC = 4 × 22 / 9 = 88 / 9
Then de = √ (BD & sup2; - be & sup2;) = √ (16-88 / 9) = 2 √ 14 / 3

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