Angle bisectors in triangles
Geometry is indeed a fascinating topic. However, many students tend to have alot of misconceptions. One type of fallacy is to assume some property of a figure by looking at the diagram. The diagram, sadly is not drawn to scale. Hence even though a picture is worth a thousand words, but it takes a mathematician to explain it completely.
Take for instance figure 1. Given that the cevian AD bisects the angle at D, is it true that it also bisects the length AC?
Some students I've taught felt that it is true. But of course, it is not. A bit of simple mathematics will prove otherwise. I'll leave this to the reader to show the following:
If AD bisects the angle at A, then the ratio AD:DC=AB:BC.