Thales Theorem

$$\frac{\overline{A_1{A}_2}}{\overline{A_2{A}_3}}=\frac{\overline{B_1{B}_2}}{\overline{B_2{B}_3}}$$

$$\frac{\overline{A_1{B}_1}}{\overline{A_2{B}_2}}=\frac{\overline{O{A}_1}}{\overline{O{A}_2}}=\frac{\overline{O{B}_1}}{\overline{O{B}_2}}$$