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

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SUMMARY

The discussion focuses on proving that in triangle ABC, where AB is the longest side, the perpendicular from point C to line AB intersects at point D, establishing the relationship A*D*B. Participants confirm that angle ACB is the largest angle due to AB being the longest side. The reasoning involves understanding that CD, the perpendicular segment, is shorter than both CB and CA, leading to contradictions if A*B*D is assumed. The conclusion emphasizes the necessity of recognizing the properties of triangles and the implications of the Pythagorean theorem in this context.

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  • Understanding of triangle properties and inequalities
  • Familiarity with the Pythagorean theorem
  • Knowledge of perpendicular lines and their geometric significance
  • Basic concepts of angle relationships in triangles
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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.

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

Duh! :rolleyes: Thanks a lot for your help. :smile:
 

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