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.
- سوريا: الصف ١١
- إيران: الصف ١١
- أوكرانيا: الصف ٨
- روسيا: الصف ٨
- ألمانيا: الصف ٩
- بريطانيا: السنة ١١
لم يُترجَم هذا المختبر إلى هذه اللغة بعد.
الدليل
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 AD٢ = AB×AC − BD×DC.