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