Circle theorems and the angle bisector theorem
Prove that two tangents from a point are equal, and learn that the bisector of an angle of a triangle cuts the opposite side in the ratio of the other two sides.
- UK: Year 11
- Germany: grade 9
- Iran: grade 11
- Syria: grade 11
- Ukraine: grade 8
- Russia: grade 8
Guide
From a point outside a circle you can draw two tangents to the circle. Dragging the point shows that the two tangents always have the same length; but measuring is not proof. For the proof we use the right-angled triangles OAP and OBP: a tangent is perpendicular to the radius and two radii are equal.
The bisector of angle A in triangle ABC cuts BC at D, and BD : DC = AB : AC. The ratio of the two parts of the opposite side equals the ratio of the two sides next to the angle; D is not the midpoint.
The length of the bisector is given by AD2 = AB×AC − BD×DC.