Cho đường tròn tâm O và điểm S nằm ngoài đường tròn tâm O. Vẽ hai tiếp tuyến SB SC với đường tròn tâm O. B; C là các tiếp điểm. Lấy điểm A thuộc cung lớn BC của đường tròn tâm O sao cho AB < AC và dây AC không đi qua O

1. Ta có SB và SC là tiếp tuyến của đường tròn tâm O nên tứ giác SBOC là tứ giác nội tiếp.

2. Ta có tứ giác SBOC là tứ giác nội tiếp nên theo định lí Ptolemy, ta có:
$$SB \cdot OC + SC \cdot OB = SO \cdot BC$$
$$\Rightarrow SE \cdot OC + SE \cdot OB = SO \cdot BC$$
$$\Rightarrow SE \cdot (OC + OB) = SO \cdot BC$$
$$\Rightarrow SE \cdot 2OE = SO \cdot BC$$
$$\Rightarrow SE = \frac{SO \cdot BC}{2OE}$$
Vậy ta có:
$$BE^2 = BO^2 + OE^2 = BO^2 + (OE + OE)^2 = BO^2 + 4OE^2 = 5OE^2$$
$$\Rightarrow OE \cdot SE = \frac{SO \cdot BC}{2} = \frac{BE^2}{5}$$
$$\Rightarrow BE^2 = OE \cdot SE$$

3. Ta có tứ giác SBOC là tứ giác nội tiếp nên góc BOC là góc ngoại tiếp của tam giác SOF nên góc BOC bằng góc SOF. Tương tự, góc BCO bằng góc SFO. Do đó, tam giác SOF đồng dạng với tam giác BOC.
Khi đó, ta có:
$$\angle BOC = \angle SOF = \angle SIF = \angle BIC$$
$$\Rightarrow BIOC \text{ nội tiếp}$$
Do đó, ta có:
$$\angle BIC = \angle BOC = \angle SIF = \angle SIB$$
$$\Rightarrow \angle BIS = \angle BIC + \angle CIS = \angle SIB + \angle CIS = \angle BIC + \angle CIS = \angle BOC + \angle CIS = \angle BOS$$
$$\Rightarrow \angle BOS = \angle BIS$$
Tương tự, ta có:
$$\angle COS = \angle CIS$$
Vậy ta có:
$$\angle BOS = \angle BIS = \angle CIS = \angle COS$$
$$\Rightarrow \angle BOS = \angle COS$$
$$\Rightarrow BS = CS$$
Khi đó, ta có:
$$\angle BSC = \angle BCS = \angle BCO = \angle BSO$$
$$\Rightarrow \angle BSO = \angle BSC$$
$$\Rightarrow BS = BO$$
Tương tự, ta có:
$$CS = CO$$
Do đó, ta có:
$$\frac{BS}{BO} = \frac{CS}{CO}$$
$$\Rightarrow \frac{BS}{SO} = \frac{CS}{SO}$$
$$\Rightarrow BS = CS$$
Khi đó, ta có:
$$\angle BSO = \angle CSO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$
$$\Rightarrow \angle BSO = \angle BCO$$