Proving A*D*B in a Triangle with AB as the Longest Side

[steelphantom]

Homework Statement

Let ABC be a triangle, and suppose that AB is the largest side. Prove that the perpendicular from C to the line AB crosses at some point D with A*D*B.

Homework Equations

The Attempt at a Solution

I know that since AB is the largest side, that angle ACB is the largest angle. I also know that CD < CB and CD < CA because a perpendicular is the shortest distance from a point to a line.

I think what I want to say is something like: If we have A*B*D, then we would have CB < CD, which is a contradiction. Is this the right line of thinking, or am I going about this the wrong way?

 

[Dick]

I think you are thinking along the right general direction. But I don't know if CD is the right segment to focus on. If the order of the points is A*B*D then don't you know AC>AD and AD>AB?

 

[steelphantom]

I think you are thinking along the right general direction. But I don't know if CD is the right segment to focus on. If the order of the points is A*B*D then don't you know AC>AD and AD>AB?

How do I know that AC > AD from that? If that's true, then I have a contradiction, since AB is supposed to be the longest side, but AC is longer than it.

 

[Dick]

AC is the hypotenuse of a right triangle and AD is a leg. AC>AD. Pythagoras told me.

 

[steelphantom]

AC is the hypotenuse of a right triangle and AD is a leg. AC>AD. Pythagoras told me.

Duh! Thanks a lot for your help.