Altitude, Distance
In an isosceles triangle ABC (AB = BC), the sum of the distances from any point on AC to the equal sides is equal to the altitude of the equal sides.
See complete Problem 223 at:
gogeometry.com/problem/p223_viviani_theorem_isosceles_triangle.htm
Level: High School, SAT Prep, College geometry
Tuesday, January 13, 2009
Elearn Geometry Problem 223: Viviani Theorem, Isosceles Triangle
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Area of ABC = Area of ABD + Area of ACD. So :
ReplyDeleteAH*BC/2= DF*AB/2 + DE*BC/2.Because AB=BC, we have :
h*AB/2=f*AB/2 + e*AB/2, from where it results that :
h=e+f
tr DEC similar to AHC
ReplyDelete=> AC/DC = h/e (1)
tr DEC similar to AFD
=> (AC-DC)/DC = f/e => AC/DC = (f+e)/e (2)
from (1) and (2)
h/e = (f+e)/e => h = f+e
thanks!
ReplyDeleteTriangle AFD , DEC and AHC are similar.
ReplyDeleteFrom the similarity of DEC and AHC we get :
e/DC = h/AC
eAC = hDC
and from the similarity of AFD and AHC we get
f/AD = h/AC
fAC = hAD
Adding these two equations and using that fact that AD+DC=AC we get:
(e+f)AC=h(AD+DC)
(e+f)AC=hAC
e+f=h
Drop Perpendicular from D to AH to meet at P => HP=e
ReplyDeleteObserve that triangles AFD and DPA are congruent(ASA)=>PA=FD=f
=> HA=HP+PA
=> h=e+f