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 2 years ago
I am trying to solve the following problem, and I have a solution which I don't understand, can you help me?
"Show that in a triangle the perpendiculars drawn from the vertices are concurrent. "
 2 years ago
I am trying to solve the following problem, and I have a solution which I don't understand, can you help me? "Show that in a triangle the perpendiculars drawn from the vertices are concurrent. "

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2bornot2b
 2 years ago
Best ResponseYou've already chosen the best response.0Here is the solution that I don't understand

phi
 2 years ago
Best ResponseYou've already chosen the best response.1We start with a given: 2 of the perpendiculars (AD and BE) meet at point O now show that the third perpendicular also meets at point O i.e. if the line segment CO extended to side AB (at point F) is perpendicular to AB then all three perpendiculars meet at O Does that make sense?

2bornot2b
 2 years ago
Best ResponseYou've already chosen the best response.0Let me read it just a sec

2bornot2b
 2 years ago
Best ResponseYou've already chosen the best response.0This line can you state it in a better way, I don't get it "if the line segment CO extended to side AB (at point F) is perpendicular to AB then all three perpendiculars meet at O "

2bornot2b
 2 years ago
Best ResponseYou've already chosen the best response.0OK, I got it... right, you are right, what next?

phi
 2 years ago
Best ResponseYou've already chosen the best response.1The 3rd perpendicular starts at vertex C. We can draw a line CO (C to O). if we continue, CO intersects the third side AB at F. Now if it turns out that COF is perpendicular to AB, then it is the perpendicular, and it goes through O, the same point as the other 2 perpendiculars

2bornot2b
 2 years ago
Best ResponseYou've already chosen the best response.0So we have to show that if the line CO is extended, and at the point where it meets AB it turns out to be perpendicular, then we are done right?

phi
 2 years ago
Best ResponseYou've already chosen the best response.1yes. And the proof relies on vectors, and the fact that the dot product of two vectors that are perpendicular = 0

2bornot2b
 2 years ago
Best ResponseYou've already chosen the best response.0What is the need of that constant l in the equation \(la.(c−b)=0\)

phi
 2 years ago
Best ResponseYou've already chosen the best response.1good question. It is obviously true, but irrelevant. I would just claim a dot (cb)= 0

phi
 2 years ago
Best ResponseYou've already chosen the best response.1Maybe they want to say that AO is too short (i.e. it does not reach the other side), but that we can scale it so that it does.

phi
 2 years ago
Best ResponseYou've already chosen the best response.1but vectors do not have to "intersect"

phi
 2 years ago
Best ResponseYou've already chosen the best response.1I assume that relabeling OA as a, etc makes sense. Do you see how BC= c  b (where all 3 are treated as vectors)?

phi
 2 years ago
Best ResponseYou've already chosen the best response.1Personally, I always "think" vector addition BC+B= c (using head to tail to add), and then rearrange the vectors to get the difference.

2bornot2b
 2 years ago
Best ResponseYou've already chosen the best response.0OK, so I will be going through the solution, and try to understand it, and if I find any problem, I will post it here. Please come again if you find a new post made on this problem :)

2bornot2b
 2 years ago
Best ResponseYou've already chosen the best response.0OK, its clear thank you! I wish I could provide you more medals. It's rare that someone digs in the unanswered questions, like you..
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