problem i encounter when trying to determine center of mass of triangle

[hyourinn]

this is my first time learn about center of mass and so far i given vector addition formula to solve this kind of question.
as most of you already know, it something like

(x1m1+x2m2)/(m1+m2) or sometimes mass is replaced by area depending on the question. and in my case i split a triangle (2D) into 3 lines(each1D) locate the center of mass of every line ( with half of it length) and calculate it with length of each line then project it to both x and y axis. so the vector addition above basically become
(x1l1+x2l2)/(l1+l2) this formula is what i found throughout the book regarding triangle case, to make it easy ill solve problem above by this equation

for x axis (x1l1+x2l2+x3l3)/(l1+l2+l3)
where 1= ED 2=EF 3=DF(1.2+1.2√2+2.2)/(2+2√2+2)=1.292
and exact same method for y axis
and give same result, and from here the problem is arises , now if we take a look 1.292 is length from left side to right (from black point 0,0 to centrium that already projected to x axis) and so it length from point D is 2-1.292=½√2 then take a look at y axis,based on this theorem distance from zero point to centrium that already projcted to y axis is 1.292 it mean from point D it has length ½√2 now if we use phytagoran theorem 「from data above we should get the distance from point D to centrium right? so√((½√2)²+(½√2)²)=1 take a look the result is one and if we applied ⅔ theorem to this we get further problem ⅔D-median = distance from point D to centriUm, then we get median from this 1/(⅔)= 1.5

even with sIngle glance it already obvious that median that correspond to point D should be √2 and not 1.5 」
so i try solve this with the ⅔ theorem,
we know D-median = ⅔√2 then project it to both x and y axis
of course the angle is 45 since that 90 is divided by half when we determine the D-median x= ⅔√2.½√2=⅔(length from D to centrium . cos45) and so y axis ⅔√2.½√2=⅔(length from D to centrium . sin45) from here we know both distance to centrium that already correspond to x and y axis so we make it related to zero point instead of point D so for x axis we change it 2-⅔= 4/3 and for y axis 2-⅔= 4/3

therr it is , but there is no single answer in book that use ⅔ theorem instead all use vector addition but if i use vector addition is give HARSH consequence (see place i marked above) so i hoping someone kInd enough to explain to me where did i go wrong ? and further about validity of this theorem ( vector addition that involve single line) because just lIke iwrite above book uses this method throughoutly and i just confused since it broke the triangle itself

 

[Dr.Peterson]

I'm having trouble following everything you wrote, but there are at least two problems here.

First, when I evaluate (1*2+1*2√2+2*2)/(2+2√2+2), I get 1.1464, not 1.292; it simplifies to (6 - √2)/4, not to 2 - √(2)/2 as you claim. Perhaps I am missing something.

Second, I think you are confusing the center of mass of a triangular lamina (that is, the interior of a triangle) with the center of mass of the edges of the triangle. See Wikipedia, which says,

The centroid is also the physical center of mass if the triangle is made from a uniform sheet of material; or if all the mass is concentrated at the three vertices, and evenly divided among them. On the other hand, if the mass is distributed along the triangle's perimeter, with uniform linear density, then the center of mass lies at the Spieker center (the incenter of the medial triangle), which does not (in general) coincide with the geometric centroid of the full triangle.
​

It is the former that is located in this case at (4/3, 4/3).

If this doesn't answer everything, please try rewriting everything with at least these two points corrected.

 

[hyourinn]

I'm having trouble following everything you wrote, but there are at least two problems here.

First, when I evaluate (1*2+1*2√2+2*2)/(2+2√2+2), I get 1.1464, not 1.292; it simplifies to (6 - √2)/4, not to 2 - √(2)/2 as you claim. Perhaps I am missing something.

Second, I think you are confusing the center of mass of a triangular lamina (that is, the interior of a triangle) with the center of mass of the edges of the triangle. See Wikipedia, which says,

The centroid is also the physical center of mass if the triangle is made from a uniform sheet of material; or if all the mass is concentrated at the three vertices, and evenly divided among them. On the other hand, if the mass is distributed along the triangle's perimeter, with uniform linear density, then the center of mass lies at the Spieker center (the incenter of the medial triangle), which does not (in general) coincide with the geometric centroid of the full triangle.
​

It is the former that is located in this case at (4/3, 4/3).

If this doesn't answer everything, please try rewriting everything with at least these two points corrected.

first thank you very much for your reply , but i still get
(1*2+1*2√2+2*2)/(2+2√2+2)= 1.292 from (6+2√2)/(4+2√2) as much i typing calculator, so would you please show me where you can got 1.1464 ?
and second ill quite doubt about vector addition in this case since like i said before it usually used whether in mass or area
so i trying surfing in google and i found thishttps://www.mathopenref.com/coordcentroid.html
if i follow this step ill exactly get result (4/3,4/3) just like i tried to solve it with ⅔ thorem, so i would like ask your opinion regarding this matter, are those vector addition ( In 1D ) is actually applicable or just something forced ( i said before, book give me this solution for most cases and it contradicting with other valid theorem

 

[Dr.Peterson]

first thank you very much for your reply , but i still get
(1*2+1*2√2+2*2)/(2+2√2+2)= 1.292 from (6+2√2)/(4+2√2) as much i typing calculator, so would you please show me where you can got 1.1464

I copied a number wrong; now I get your answer, and it does simplify to 2 - sqrt(2)/2.

and second ill quite doubt about vector addition in this case since like i said before it usually used whether in mass or area
so i trying surfing in google and i found thishttps://www.mathopenref.com/coordcentroid.html
if i follow this step ill exactly get result (4/3,4/3) just like i tried to solve it with ⅔ thorem, so i would like ask your opinion regarding this matter, are those vector addition ( In 1D ) is actually applicable or just something forced ( i said before, book give me this solution for most cases and it contradicting with other valid theorem

I answered this before. The center of mass of the three edges is NOT the same as the center of mass (centroid) of the triangle (including the interior). Both are valid concepts, but they are two different things.