Triangle Proof: Parallel Sides and ProportionalityDate: 07/27/2004 at 09:24:25 From: Sue Subject: Ratio How do I prove that a line which cuts two sides of a triangle proportionately is parallel to the third side? I can prove that a straight line drawn parallel to one side of a triangle will divide the other two sides proportionately, but I really have no idea how to prove the converse. Date: 07/28/2004 at 09:15:09 From: Doctor Willae Subject: Re: Ratio Sue, Thanks for writing to Dr. Math. When dealing with geometric proofs like this, I like to draw things out. And, when I do, I like to make it as simple as possible. So, for example, when I draw a triangle for your problem, I'm going to put one vertex at the origin and another on the x-axis. Let's take a look (and please forgive the bad ASCII art): (B, C) * * * * * * * * * * * * * * * * * * * * * *********************** (0, 0) (A, 0) OK, here's what's going on. As I said, I've placed one vertex at the origin (0, 0). I put the second vertex along the x-axis at some point (A, 0). I can't make any assumptions about the third vertex without losing generality, so it's just at some point, (B, C). Now let's add in the cutting line. Let P be the fraction of the side that's cut, as measured from point (B,C). (This implies that P must be in the range [0, 1].) (B, C) * * * * * * * * * * * * * * * * * (P * B, P * C) +++++++++++++++++++ (A + P * (B - A), P * C) * * *********************** (0, 0) (A, 0) All that remains to show is that the slope of the cutting line matches the slope of the third side. They're both zero so you're good to go. Let me know if you have any more questions about this. - Doctor Willae, The Math Forum http://mathforum.org/dr.math/ Date: 07/30/2004 at 22:51:03 From: Sue Subject: Thank you (Ratio) Doctor Willae - Thank you very much for your help. I really appreciate it. |
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