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 Triangular Polygons (Posted on 2011-08-12)
Three points have been chosen randomly from the vertices of a n-sided regular polygon.

Determine the probability (in terms of n) that they form (a) an acute triangle; (b) a right triangle?

 See The Solution Submitted by K Sengupta No Rating

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 computer findings for triangles through 50-gons | Comment 1 of 4
` n  p(acute)     p(right) 3 1.000000000 0.00000000 4 0.000000000 1.00000000 5 0.666666667 0.00000000 6 0.200000000 0.60000000 7 0.600000000 0.00000000 8 0.285714286 0.42857143 9 0.571428571 0.0000000010 0.333333333 0.3333333311 0.555555556 0.0000000012 0.363636364 0.2727272713 0.545454545 0.0000000014 0.384615385 0.2307692315 0.538461538 0.0000000016 0.400000000 0.2000000017 0.533333333 0.0000000018 0.411764706 0.1764705919 0.529411765 0.0000000020 0.421052632 0.1578947421 0.526315789 0.0000000022 0.428571429 0.1428571423 0.523809524 0.0000000024 0.434782609 0.1304347825 0.521739130 0.0000000026 0.440000000 0.1200000027 0.520000000 0.0000000028 0.444444444 0.1111111129 0.518518519 0.0000000030 0.448275862 0.1034482831 0.517241379 0.0000000032 0.451612903 0.0967741933 0.516129032 0.0000000034 0.454545455 0.0909090935 0.515151515 0.0000000036 0.457142857 0.0857142937 0.514285714 0.0000000038 0.459459459 0.0810810839 0.513513514 0.0000000040 0.461538462 0.0769230841 0.512820513 0.0000000042 0.463414634 0.0731707343 0.512195122 0.0000000044 0.465116279 0.0697674445 0.511627907 0.0000000046 0.466666667 0.0666666747 0.511111111 0.0000000048 0.468085106 0.0638297949 0.510638298 0.0000000050 0.469387755 0.06122449`

DEFDBL A-Z
CLS
FOR n = 3 TO 50
trict = 0: rtct = 0: obtct = 0
FOR s1 = 1 TO n - 1
FOR s2 = 1 TO n - s1 - 1
c1 = s1 * 360 / n
c2 = s2 * 360 / n
c3 = (n - s1 - s2) * 360 / n
a1 = 180 - (c1 + c2) / 2
a2 = 180 - (c2 + c3) / 2
a3 = 180 - (c3 + c1) / 2
trict = trict + 1
IF ABS(a1 - 90) < .00000001# OR ABS(a2 - 90) < .00000001# OR ABS(a3 - 90) < .00000001# THEN
rtct = rtct + 1
END IF
IF a1 > 90.00000000999999# OR a2 > 90.00000000999999# OR a4 > 90.00000000999999# THEN
obtct = obtct + 1
END IF
NEXT s2
NEXT
PRINT USING "### #.######### #.########"; n; (trict - obtct - rtct) / trict; rtct / trict
NEXT n

 Posted by Charlie on 2011-08-14 16:58:33

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