Theory of graphs by Oystein Ore

By Oystein Ore

Quite a few difficulties were incorporated. a lot of those are very simple; others are extra within the nature of proposed examine difficulties.

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Nb C b a d A B m AB = c D n Show that for any ABC as shown in the figure above the relationship between the lengths of the labeled line segments is given by a2 m + b2 n = c(d2 + mn). —– Collinear Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . nb Show that the points P1 (r1 , θ1 ), P2 (r2 , θ2 ) and P3 (r3 , θ3 ) in polar coordinates are collinear if and only if −r1 r2 sin(θ1 − θ2 ) + r1 r3 sin(θ1 − θ3 ) − r2 r3 sin(θ2 − θ3 ) = 0.

Nb C b a d A B m AB = c D n Show that for any ABC as shown in the figure above the relationship between the lengths of the labeled line segments is given by a2 m + b2 n = c(d2 + mn). —– Collinear Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . nb Show that the points P1 (r1 , θ1 ), P2 (r2 , θ2 ) and P3 (r3 , θ3 ) in polar coordinates are collinear if and only if −r1 r2 sin(θ1 − θ2 ) + r1 r3 sin(θ1 − θ3 ) − r2 r3 sin(θ2 − θ3 ) = 0.

Solving this equation for x yields x1 r2 + x2 r1 . 1) x= r1 + r2 Similarly, y1 r2 + y2 r1 . 2) specialize to x= y1 + y2 x1 + x2 and y = . 3) also have useful physical interpretations. 2), let x and y be the coordinates of the center of gravity of masses r1 and r2 placed at P1 and P2 , respectively. 3). It is of further interest to note the positions of P for various values of the ratio r1 /r2 . If this ratio is zero, then P coincides with P1 , and if this ratio is a positive number, P is an internal point of division.

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