MaplePrimes - answers and comments on Question, Coordinates of the center circle circumscribed triangle are integer numbers

Solution scheme

Kitonum — Thu, 30 Aug 2012 21:14:08 Z

You can build any number of triangles for which the coordinates of the vertices and the coordinates of the center of circle circumscribed are integers as follows:

1) Take any three points with integer coordinates that do not lie on a straight line. Let be points A, B, C.

2) Find the coordinates of the center of circle circumscribed about triangle ABC as the point of intersection of three planes. The coordinates of this point will be rational numbers.

3) Find the least common multiple of the denominators of the fractions.

4) Multiply the coordinates of all the points on this number.

Thank you very much.

toandhsp — Fri, 31 Aug 2012 04:03:41 Z

> restart;with(geom3d):

point(A,2,8,2):

point(B,4,-8,12):

point(C,2,6,4):point(M,a,b,c):

eq1:=distance(A,M) = distance(B,M):

eq2:=distance(A,M) = distance(C,M):

eq3:=Equation(plane(ABC,[A,B,C],[a,b,c])):

sys:=solve([eq1,eq2,eq3],[a,b,c]):

coordinates(point(M,eval(coordinates(M), op(sys))));

triangle(ABC,[A,B,C]):

IsEquilateral(ABC);

IsRightTriangle(ABC);

Another way

toandhsp — Fri, 31 Aug 2012 18:54:58 Z

> restart;with(geom3d):

point(A,1,-4,3):

point(B,3,-2,3):

point(C,2, 1, 1):point(M,a,b,c):

eq1:=distance(A,M) = distance(B,M):

eq2:=distance(A,M) = distance(C,M):

eq3:=Equation(plane(ABC,[A,B,C],[a,b,c])):

sys:=solve([eq1,eq2,eq3],[a,b,c]):

coordinates(point(M,eval(coordinates(M), op(sys))));

triangle(ABC,[A,B,C]):

IsEquilateral(ABC);

IsRightTriangle(ABC);

k:=lcm(denom(xcoord(M)),denom(ycoord(M)),denom(zcoord(M))):

point(o,0,0,0):

homothety(T, ABC, k, o):

map(coordinates,DefinedAs(T));

coordinates(homothety(H, M, k, o));