In an equilateral triangle, prove that the sum of the distances from any interior point to the sides is equal to the altitude of the triangle.
See complete Problem 221 at:
gogeometry.com/problem/p221_viviani_theorem_equilateral_triangle.htm
Level: High School, SAT Prep, College geometry
Tr.ABC = Tr.BDA + Tr.ADC + Tr.CDB
ReplyDeleteAC*h/2 = BA*f/2 + AC*g/2 + CB*e/2
But AC=CB=BA since ABC is equilateral. Hence h=f+g+e
QED
Ajit: ajitathle@gmail.com
area of ABC = (1/2)*h*AC
ReplyDeletealso area of ABC = area of ADC + area of BDA + area of BDC = (1/2)*g*AC + (1/2)*f*AB + (1/2)*e*BC.
which implies that h*AC = g*AC + f*AB + e*BC
but AB = AC = BC since ABC is equilateral.
therefore h = e + f + g.
http://www.youtube.com/watch?v=0AOqIfTxCDg
ReplyDeleteDraw a line through point D parallel too the line AC. The line meets the line AB in M and BC in N.
ReplyDeleteThe line BH meets the line MN in the point Q.
BQ=h-g
Draw an altitude from point M onto the line BN. The altitude meets the line BN on the point P.Because the triangle BMN is equilateral we have:
MP=BQ=h-g
Also from problem 223 we have:
MP=e+f
Therefore
e+f=h-g
e+f+g=h