Tuesday, January 13, 2009

Elearn Geometry Problem 223: Viviani Theorem, Isosceles Triangle

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.

 Geometry Problem 223. Viviani Theorem
See complete Problem 223 at:

Level: High School, SAT Prep, College geometry


  1. Area of ABC = Area of ABD + Area of ACD. So :
    AH*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 :

  2. tr DEC similar to AHC

    => 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

  3. Triangle AFD , DEC and AHC are similar.

    From 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:


  4. Drop Perpendicular from D to AH to meet at P => HP=e
    Observe that triangles AFD and DPA are congruent(ASA)=>PA=FD=f
    => HA=HP+PA
    => h=e+f