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The alpha-version of the Stewart's theorem

Gelişgen Ö., Kaya R.

Demonstratio Mathematica

2013

Vol. 46, nr 4

795--808

G. Chen [1] developed Chinese checker metric for the plane on the question “how to develop a metric which would be similar to the movement made by playing Chinese checker” by E. F. Krause [13]. Tian [17] developed α-metric which is defined by dα(P1,P2)=max {|x1 - x2|, |y1 - y2|} + (sec α - tanα) min {|x1 - x2|, |y1 - y2|} where P1= (x1, y1) and P2= (x2, y2) are two points in analytical plane, and α ϵ [0, π/4]. Stewart’s theorem yields a relation between lengths of the sides of a triangle and the length of a cevian of the triangle. A taxicab and Chinese checkers analogues of Stewart’s theorem are given in [12] and [9], respectively. In this work, we give an α-analog of the theorem of Stewart by using the base line concept and we give α-analog of formulae for the medians which is the application of Stewart’s theorem.