Art of Problem Solving: 2012 AMC 10 B #23

[@abhichakladar5279]

Bring in Harvey. He could see something awesome that could solve the problem!

[@זאבגלברד]

If you know how to deal with 3D coord you find the plain equation that includes (0,0,0) , (1,0,1) , and (1,1,0) and then find distance from (0,1,1) to that plain - something very technical .

[@117117nobody]

i didn't get the third method tho... could you explain a little more

[@flamingflameblade5760]

For the second solution, how would you find the distance between the two points after you determined that the middle of the triangle was 1/3,1/3,1/3?

[@amritlohia8240]

Once you have the coordinates as (1/3,1/3,1/3), you can see it's 1/3 of the way from (0,0,0) to (1,1,1), so the distance is 1/3 of the entire diagonal. Alternatively, you can use the distance formula: the distance from (a,b,c) to (d,e,f) is sqrt((a-d)^2+(b-e)^2+(c-f)^2) (it's just a 3D version of Pythagoras' theorem).

[@anupamghosh3056]

The last solution was awesome.

[@RinniBhansali]

lol u forgot to write the r in area at 5:53 hehehehe :) aea

[@M_Chen333]

@Rinni Bhansali No, he wrote Aeq, meaning area of equilateral triangle.