There's a pentagon, not necessarily regular, with integral sides, and a point within it from which lines are drawn to each of the pentagon's vertices. Each of these five lines from the internal point is a different integral length less than 25, and their angles with each other at that internal point are an integral number of degrees, none of which is 90.
How long are the lines and the sides of the pentagon? What is its total perimeter?
I used the following Mathematica code
Side[x_,y_,z_]:=Sqrt[x^2+y^2-2*x*y*Cos[z*Pi/180]];
For[s1=1,s1<25,s1++,
For[s2=1,s2<25,s2++,
If[s2!=s1,
For[t1=1,t1<180,t1++,
If[t1„j90,
a1=Side[s1,s2,t1];
If[IntegerQ[a1],
For[s3=1,s3<25,s3++,
If[s3„js1 && s3„js2,
For[t2=1,t2<Min[180,360-t1-2],t2++,
If[t2„j90,
a2=Side[s2,s3,t2];
If[IntegerQ[a2],
For[s4=1,s4<25,s4++,
If[s4„js1 && s4„js2 && s4„js3,
For[t3=1,t3<Min[180,360-t1-t2-1],t3++,
If[t3„j90,
a3=Side[s3,s4,t3];
If[IntegerQ[a3],
For[s5=1,s5<25,s5++,
If[s5„js1 && s5„js2 && s5„js3 && s5„js4,
For[t4=1,t4<Min[180,360-t1-t2-t3],t4++,
t5=360-t1-t2-t3-t4;
a4=Side[s4,s5,t4];
a5=Side[s5,s1,t5];
If[IntegerQ[a4] && IntegerQ[a5],
Print["Sides: ",a1," ",a2," ",a3," ",a4," ",a5];
Print["Distances: ",s1," ",s2," ",s3," ",s4," ",s5];
Print["Angles: ",t1," ",t2," ",t3," ",t4," ",t5];
];];];];];];];];];];];];];];];];];];];];
and got the sole solution of
Line lengths: 5,8,15,24,21
Side Lengths: 7,13,21,39,19
Angles: 60,60,60,120,60
for a total perimeter of 99
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Posted by Daniel
on 2009-06-09 13:48:33 |