The three vertices of an equilateral triangle are at distances 1, 2, and 3 from a line.
Find all possible areas of this triangle.
Using Geometer's Sketchpad, when the distance2 vertex is the sole point on its side of the line the area is about 10.9, but if the distance3 is the isolated vertex the area is about 12.0. If the distance1 vertex is the one isolated, the area is about 6.93 or 7,5 depending on the construction.
(The 6.93 is spurious; see below).
If all are on the same side, I get 1.73 as the area.
I suspect the other cases have alternatives also, but it's hard to see the appropriate construction. (again, see below  the Matlab calculation)
Matlab calculation:
syms x1 x2 x3
for y1=[1 1]
for y2=[2 2]
for y3=[3 3]
x = vpasolve([(x1x2)^2+(y1y2)^2==(x2x3)^2+(y2y3)^2, ...
(x1x2)^2+(y1y2)^2==(x1x3)^2+(y1y3)^2, ...
abs(atand((y3y1)/(x3x1))atand((y2y1)/(x2x1)))==60], ...
[x1,x2,x3]);
if ~isempty(x.x1)
fprintf('(%9.6f,%2d) (%9.6f,%2d) (%9.6f,%2d) %9.6f %9.6f
', ...
x.x1,y1,x.x2,y2,x.x3,y3, ...
(sqrt(3)/4)*((x.x1x.x2)^2+(y1y2)^2), ...
(sqrt(3)/4)*((x.x1x.x3)^2+(y1y3)^2))
end
end
end
end
finds
vertices area based on
side 1 side 2
(40.716919, 1) (45.913071, 2) (44.181020,3) 12.124356 12.124356
(25.356636, 1) (21.315184,2) (20.737834, 3) 10.969655 10.969655
(25.356636,1) (21.315184, 2) (20.737834,3) 10.969655 10.969655
(40.716919,1) (45.913071,2) (44.181020, 3) 12.124356 12.124356
which seem to be the 10.9 and 12.1 found via GSP. The limitation would be arising from looking for a difference of 60.
Changing the sought difference in angles to 120° gives
(43.026320, 1) (45.913071,2) (41.871619,3) 7.505553 7.505553
(43.026320,1) (45.913071, 2) (41.871619, 3) 7.505553 7.505553
which is the 7.5 found by GSP.
specifying "digits 10" before doing the vpasolve gives, when looking for 60°,
(38.802870, 1) (37.070819, 2) (38.802870, 3) 1.732051 1.732051
(40.716919, 1) (45.913071, 2) (44.181020,3) 12.124356 12.124356
(25.356636, 1) (21.315184,2) (20.737834, 3) 10.969655 10.969655
(25.356636,1) (21.315184, 2) (20.737834,3) 10.969655 10.969655
(40.716919,1) (45.913071,2) (44.181020, 3) 12.124356 12.124356
(38.802870,1) (37.070819,2) (38.802870,3) 1.732051 1.732051
now finding the 1.7 solution from GSP.
Using angle difference of 120° with digits 10:
(43.026320, 1) (44.758370, 2) (43.026320, 3) 1.732051 1.732051
(43.026320, 1) (45.913071,2) (41.871619,3) 7.505554 7.505553
(43.026320,1) (45.913071, 2) (41.871619, 3) 7.505554 7.505553
(43.026320,1) (44.758370,2) (43.026320,3) 1.732051 1.732051
The Matlab program has found
1.732051
7.505553
10.969655
12.124356
What about GSP's 6.93?
Going back, I see that I mishandled the case where the area was 6.93. I confused two lines and the distances of the vertices from the line were 1, 1, and 3 rather than 1, 2 and 3.
Edited on March 29, 2024, 1:31 pm

Posted by Charlie
on 20240329 13:30:11 