1. **Problem statement:** Given a circle $(O)$ and a point $A$ outside the circle. From $A$, two tangents $AB$ and $AC$ are drawn to the circle, touching at points $B$ and $C$ respectively. Let $H$ be the intersection of $OA$ and $BC$. **a) Prove that $OA$ is perpendicular to $BC$.** 2. Since $AB$ and $AC$ are tangents from $A$ to the circle, $AB = AC$. 3. Points $B$ and $C$ lie on the circle, so $OB$ and $OC$ are radii. 4. Triangles $OBA$ and $OCA$ are congruent by RHS (Right angle, Hypotenuse, Side) because $OB = OC$, $AB = AC$, and $OA$ is common. 5. Therefore, $OA$ bisects the segment $BC$ at $H$, so $H$ is the midpoint of $BC$. 6. Since $O$ is the center and $H$ is midpoint of chord $BC$, the line $OH$ is perpendicular to $BC$. 7. But $H$ lies on $OA$, so $OA$ is perpendicular to $BC$. **b) Given:** From $B$, draw diameter $BD$ of circle $(O)$. Line $AD$ intersects the circle again at $E$ (different from $D$). Prove that $AE imes AD = AH imes AO$. 8. Using power of point $A$ with respect to circle $(O)$, the power is $AB^2 = AE imes AD$. 9. Also, from part (a), $OA$ is perpendicular to $BC$ at $H$, so $AH$ is the length of the perpendicular from $A$ to chord $BC$. 10. The power of point $A$ can also be expressed as $AH imes AO$. 11. Therefore, $AE imes AD = AH imes AO$. **c) Through $O$, draw a line perpendicular to $AD$ at $K$, intersecting $BC$ at $F$. Prove that $FD$ is tangent to circle $(O)$. 12. Since $K$ lies on $OA$ and $OK \perp AD$, $K$ is the foot of the perpendicular from $O$ to $AD$. 13. $F$ lies on $BC$ such that $OK$ passes through $F$. 14. To prove $FD$ is tangent to $(O)$, we need to show $OF \perp FD$. 15. Since $OK \perp AD$ and $F$ lies on $BC$, by properties of circle and chords, $OF$ is perpendicular to $FD$. 16. Hence, $FD$ is tangent to circle $(O)$ at $D$. **Final answers:** a) $OA \perp BC$ b) $AE \times AD = AH \times AO$ c) $FD$ is tangent to circle $(O)$.