There is a unique
tangential isosceles trapezoid whose three different side lengths and diagonal are consecutive whole numbers.
Find the ratio of circumradius to inradius for this trapezoid.
Per the diagram in Wikipedia, the smaller base is on the bottom and called b and is the smallest of the four integers. The side is next in size and the top is the largest of the givens. The diagonal needs to be the largest of all four integers.
The program calculates the diagonal from the base, the side and the top. Other formulae go on from there to get the inRadius and circumRadius.
clearvars,clc
for b=1:20
s=b+1; t=s+1; % d=t+1;
h=sqrt(b*t); inRad=h/2;
bottomAngle=90+atand((t-b)/(2*h));
dsq= s^2+b^2-2*s*b*cosd(bottomAngle) ;
d=sqrt(dsq);
circumR1=s*b*d/sqrt((s+b+d)*(s+b-d)*(s+d-b)*(b+d-s));
% just a check to see they match for the two equal diameters,
% using triangle containing top or bottom, second calc:
circumR2=s*t*d/sqrt((s+t+d)*(s+t-d)*(s+d-t)*(t+d-s));
% disp([b s t d])
% disp([circumR1 circumR2]);
% disp(' ')
fprintf('%3d %3d %3d %15.12f %15.12f %15.12f %15.12f\n' ,b,s,t,d,circumR2,inRad,circumR2/inRad);
end
base side top diagonal circumradius inradius ratio
1 2 3 2.645751311065 1.527525231652 0.866025403784 1.763834207376
2 3 4 4.123105625618 2.186606960567 1.414213562373 1.546164609607
3 4 5 5.567764362830 2.875181153713 1.936491673104 1.484737163421
* 4 5 6 7.000000000000 3.572172541559 2.449489742783 1.458333333333 ***
5 6 7 8.426149773176 4.272837799068 2.958039891550 1.444482818259
6 7 8 9.848857801796 4.975460615193 3.464101615138 1.436291762762
7 8 9 11.269427669585 5.679257720334 3.968626966597 1.431038434233
8 9 10 12.688577540450 6.383817431600 4.472135955000 1.427464973301
9 10 11 14.106735979666 7.088901554721 4.974937185533 1.424922826229
10 11 12 15.524174696260 7.794362278810 5.477225575052 1.423049347157
11 12 13 16.941074346097 8.500102837716 5.979130371551 1.421628616456
12 13 14 18.357559750686 9.206057141645 6.480740698408 1.420525456898
13 14 15 19.773719933285 9.912178471521 6.982120021884 1.419651687518
14 15 16 21.189620100417 10.618432844419 7.483314773548 1.418947774582
15 16 17 22.605309110915 11.324794938195 7.984359711336 1.418372336371
16 17 18 24.020824298929 12.031245490620 8.485281374239 1.417895878756
17 18 19 25.436194683954 12.737769584735 8.986100377806 1.417496917372
18 19 20 26.851443164195 13.444355486391 9.486832980505 1.417159500333
19 20 21 28.266588050205 14.150993836703 9.987492177719 1.416871581464
20 21 22 29.681644159312 14.857677078932 10.488088481702 1.416623925785
As the numbers get larger, the figure approaches being a square.
The appropriate result is the line where the base is 4, the side is 5, the top is 6.
The resulting diagonal is 7.000000000000 and the circumradius is 3.572172541559 (35/(4*sqrt(6)) per Wolfram Alpha) and the inradius is 2.449489742783 (sqrt(6) per Wolfram Alpha). Their ratio is found to be 1.458333333333, which is presumably 4.375/3 = 35/24, agreeing with Wolfram's identifications.
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Posted by Charlie
on 2022-11-30 09:39:15 |
computer solution
