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A simple geometrical theorem I did not know

EWD1226 Image 01

Given:

        AE = α•AC
        ED = β•EB
        FD = γ•FC

Denoting with (PQR) the area of ΔPQR, we observe

(ADB) = γ•(ACB)                 and also

(ABD) = (1–β)•(AEB)
           = (1–β)•α•(ACB)

from which we conclude   γ = (1–β)•α     .

[ The theorem we used thrice—say (PQR)•RS/QR = (PRS)—is no more than adding metric to —see EWD1221b—

EWD1226 Image 02

     

   R ≠ S ∧ col.R.S.Q ∧ tri.R.Q.P ⇒ tri.R.S.P ]
The theorem proved in this note is of no importance; it is recorded here because I don't remember this proof technique from my school-days.

Nuenen, 22 December 1995

prof. dr. Edsger W. Dijkstra
Department of Computer Sciences
The University of Texas at Austin
Austin, TX 78712-1188

transcribed by Swarup Sahoo
last revised Sun, 26 Jun 2011