A simple geometrical theorem I did not know
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—
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