Triangle altitude's concurrency by symmetry

[alijan kk]

Homework Statement

in the proof of triangle altitudes concurrency , i have found the equation of the Altitude AD,
x(x₂-x₃)+y(y₂-y₃)-x₁(x₂-x_3)-y₁(y₂-y₃)

Homework Equations

The Attempt at a Solution

In the book other altitude equations are written by symmetry,

how is the idea of symmetry is used here ? , .

 

[FactChecker]

The first thing you must learn is to be more careful. The problem statement looks incomplete. What is D, x, y, x1, x2, x3, y1, y2, y3?

Did the book say "symmetry"? I don't see any symmetry, but you may have left out some important parts of the problem statement. Otherwise, "symmetry" is probably the wrong word to use. A better word might be "similarly".

 

[alijan kk]

The first thing you must learn is to be more careful. The problem statement looks incomplete. What is D, x, y, x1, x2, x3, y1, y2, y3?

Did the book say "symmetry"? I don't see any symmetry, but you may have left out some important parts of the problem statement. Otherwise, "symmetry" is probably the wrong word to use. A better word might be "similarly".

i have now uploaded the image of triangle ,

 

[SammyS]

Homework Statement

in the proof of triangle altitudes concurrency , i have found the equation of the Altitude AD,
x(x₂-x₃)+y(y₂-y₃)-x₁(x₂-x₃)-y₁(y₂-y₃)
View attachment 219067

Homework Equations

The Attempt at a Solution

In the book other altitude equations are written by symmetry,

how is the idea of symmetry is used here ? , .

x(x₂-x₃)+y(y₂-y₃)-x₁(x₂-x₃)-y₁(y₂-y₃)

is a mathematical expression, not an equation. It has no equal sign.

Did you mean to set this expression equal to zero ?

 

[FactChecker]

The diagram I see now is better. Did the book say anything else about the triangle? Is it equilateral or isosceles?

 

[LCKurtz]

@alijan kk: As others have pointed out, it isn't "symmetry", but here is what your author is using. In your figure, if you rotate the labels counterclockwise you will have replaced the subscripts 1 by 2, 2 by 3, and 3 by 1.Then the same steps on the points A, B, and C that were used to get your original expression will give an answer you can get by replacing 1 by 2, 2 by 3, and 3 by 1 in your original expression. Then rotate the subscripts one more time for the third equation. (It has already been pointed out to you that you haven't written equations).

 

[SammyS]

At this point, I think we need to wait for OP to return to this thread.

 

[alijan kk]

@alijan kk: As others have pointed out, it isn't "symmetry", but here is what your author is using. In your figure, if you rotate the labels counterclockwise you will have replaced the subscripts 1 by 2, 2 by 3, and 3 by 1.Then the same steps on the points A, B, and C that were used to get your original expression will give an answer you can get by replacing 1 by 2, 2 by 3, and 3 by 1 in your original expression. Then rotate the subscripts one more time for the third equation. (It has already been pointed out to you that you haven't written equations).

VERY HELPFULL <3 yes I would take care of the description next time

 

[alijan kk]

The diagram I see now is better. Did the book say anything else about the triangle? Is it equilateral or isosceles?

No just a triangle to proov triangle alititudes concurrency