Property 1.4

I: Incentro ;  O: Circuncentro
r: Inradio ;  R: Circunradio
OM, ON y OQ : Mediatrices
ON' = ha ; OQ' = hb ; OM' = hc

• QUEREMOS HALLAR EL VALOR DE  ha+hb+hc

• Se traza NR (R en proyección de AC) tal que <NRA = 90
   Se traza OR' // AC

• <M'OA = <R'NO = <C  y  OA = ON = R
  =>  ∆M'OA = ∆R'NO ->  NR' = hc  y  OR' = c/2

• ∆NRA ~ ∆NCN' ( ya que <NCN' = <NAC = <A/2 )
  => NR/NN' = AR/CN' -> (NR'+R'R)/(ON-ON') = (AQ'+Q'R)/CN'

        (hc+hb)/(R-ha) = (b+c)/a

• a(hb+hc) = (b+c)(R-ha) ..... [1] ; Similarmente:

        b(ha+hc) = (a+c)(R-hb) ..... [2]
        c(ha+hb) = (a+b)(R-hc) ..... [3]

• Sumando [1], [2], [3] y simplificando

        a(hb+hc)+b(ha+hc)+c(ha+hb) = (a+b+c)R

• Agregando a ambos lados: aha+bhb+chc = 2(Area ∆ABC)

        (a+b+c)(ha+hb+hc) = (a+b+c)R + 2[(a+b+c)/2]r

• Finalmente:    ha+hb+hc = R+r