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A352484 Decimal expansion of the probability that when three real numbers are chosen at random, uniformly and independently in the interval [0,1], they can be the lengths of the sides of a triangle whose altitudes are also the sides of some triangle.

%I A352484 #12 Feb 16 2025 08:34:03
%S A352484 3,0,5,8,3,6,7,2,2,2,5,0,7,8,8,8,7,5,6,3,4,3,5,9,5,8,1,7,0,1,9,7,8,1,
%T A352484 7,2,1,6,0,3,2,2,4,2,0,1,4,3,4,2,6,6,0,6,7,8,3,8,7,5,0,5,8,6,0,1,1,9,
%U A352484 9,0,4,5,9,0,4,0,4,3,4,3,2,6,8,0,5,0,0,5,9,1,5,5,7,9,9,9,2,8,7,6,0,4,7,8,5
%N A352484 Decimal expansion of the probability that when three real numbers are chosen at random, uniformly and independently in the interval [0,1], they can be the lengths of the sides of a triangle whose altitudes are also the sides of some triangle.
%C A352484 Without the condition on the altitudes the probability is 1/2.
%H A352484 Mohammed Yaseen, <a href="/A352484/b352484.txt">Table of n, a(n) for n = 0..10000</a>
%H A352484 Murray S. Klamkin, <a href="https://cms.math.ca/publications/crux/issue?volume=15&amp;issue=10">Problem 1494</a>, Crux Mathematicorum, Vol. 15, No. 10 (1989), p. 298; <a href="https://cms.math.ca/publications/crux/issue?volume=17&amp;issue=2">Solution to Problem 1494</a>, by P. Penning, ibid., Vol. 17, No. 2 (1991), pp. 53-54.
%H A352484 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Altitude.html">Altitude</a>.
%H A352484 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A352484 Equals 2*log(sqrt(5)-1) + 1 - sqrt(5)/2.
%e A352484 0.30583672225078887563435958170197817216032242014342...
%t A352484 RealDigits[2*Log[Sqrt[5] - 1] + 1 - Sqrt[5]/2, 10, 100][[1]]
%o A352484 (PARI) 2*log(sqrt(5)-1) + 1 - sqrt(5)/2 \\ _Charles R Greathouse IV_, Nov 26 2024
%Y A352484 Cf. A020837, A134972, A352485.
%K A352484 nonn,cons
%O A352484 0,1
%A A352484 _Amiram Eldar_, Mar 18 2022