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

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Homework Help Overview

The problem involves proving a relationship in triangle ABC, where AB is the longest side. The original poster seeks to establish that the perpendicular from point C to line AB intersects at point D, leading to the conclusion A*D*B.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • The original poster considers the implications of AB being the longest side and explores the relationship between segments CD, CA, and CB. Some participants question the focus on segment CD and suggest examining the implications of the order of points A*B*D.

Discussion Status

Participants are actively engaging with the original poster's reasoning, offering insights into the relationships between the sides of the triangle. There is a recognition of contradictions arising from the assumptions made, particularly regarding the lengths of segments AC and AD.

Contextual Notes

There is an emphasis on the properties of right triangles and the implications of the triangle inequality, as well as the need to clarify the relationships between the segments involved in the proof.

steelphantom
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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.

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?
 
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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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