In
A Tangential Trapezoid, Jer describes a specific isosceles tangential trapezoid. I am going to generalize that idea.
How many isosceles tangential trapezoids exist where all their sides and diagonals are whole numbers and all those dimensions are at most 1000?
An example is a trapezoid with bases 6 and 20, lateral sides of 13, and diagonals of 17.
Bonus challenge: How many of these trapezoids have all dimensions at most 100,000?
clearvars,clc
ct=0;
for b=1:20000
for t=b:20000
h=sqrt(b*t); s=(t-b)/(2*sin(atan((t-b)/(2*h))));
if abs(round(s)-s)<s/1e12
bottomAngle=90+atand((t-b)/(2*h));
dsq= s^2+b^2-2*s*b*cosd(bottomAngle) ;
d=sqrt(dsq);
if abs(round(d)-d)<d/1e12
tot=t+b+2*(s+d);
if tot<=100000
fprintf('%8d %8s %8d %8s %10d\n' ,b,sym(s),t,sym(d),round(tot));
ct=ct+1;
end
end
end
end
end
ct
finds 17044 such trapezoids for the Bonus challenge.
I've pasted below the end of the prints, if anyone cares to check the accuracy of the sides and diagonals being exact integers.
base side top diagonal total
13950 14125 14300 19975 96450
13972 14471 14970 20459 98802
13986 14625 15264 20673 99846
14000 14500 15000 20500 99000
14028 14529 15030 20541 99198
14040 14316 14592 20244 97752
14040 14346 14652 20286 97956
14042 14371 14700 20321 98126
14056 14558 15060 20582 99396
14080 14216 14352 20104 97072
14084 14587 15090 20623 99594
14112 14196 14280 20076 96936
14112 14616 15120 20664 99792
14130 14569 15008 20599 99474
14140 14645 15150 20705 99990
14208 14475 14742 20469 98838
14208 14530 14852 20546 99212
14212 14450 14688 20434 98668
14280 14365 14450 20315 98090
14430 14437 14444 20417 98582
14448 14534 14620 20554 99244
ct =
17044
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Posted by Charlie
on 2022-12-16 10:51:12 |
Bonus
