Question

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. "

Answer 1

Here is the solution that I don't understand

Answer 2

We 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?

Answer 3

Let me read it just a sec

Answer 4

This 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
"

Answer 5

OK, I got it... right, you are right, what next?

Answer 6

The 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

Answer 7

So 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?

Answer 8

yes.
And the proof relies on vectors, and the fact that the dot product of two vectors that are perpendicular = 0

Answer 9

What is the need of that constant l in the equation \(la.(c−b)=0\)

Answer 10

good question. It is obviously true, but irrelevant. I would just claim
a dot (c-b)= 0

Answer 11

Maybe 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.

Answer 12

but vectors do not have to "intersect"

Answer 13

OK

Answer 14

I assume that relabeling OA as a, etc makes sense.
Do you see how BC= c - b (where all 3 are treated as vectors)?

Answer 15

Personally, I always "think" vector addition BC+B= c (using head to tail to add), and then rearrange the vectors to get the difference.

Answer 16

*b (not B)

Answer 17

OK, 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 :)

Answer 18

roger that.

Answer 19

Thanks a lot

Answer 20

OK, its clear thank you! I wish I could provide you more medals. It's rare that someone digs in the unanswered questions, like you..