theorem Th51: :: EUCLID12:70

for a being Real

for A, B, C, D being Point of (TOP-REAL 2)

for b, r being Real st A,B,C is_a_triangle & A in circle (a,b,r) & B in circle (a,b,r) & C in circle (a,b,r) & A,B,D is_a_triangle & A in circle (a,b,r) & B in circle (a,b,r) & D in circle (a,b,r) & C <> D & not the_diameter_of_the_circumcircle (A,B,C) = the_diameter_of_the_circumcircle (D,B,C) holds

the_diameter_of_the_circumcircle (A,B,C) = - (the_diameter_of_the_circumcircle (D,B,C))

for A, B, C, D being Point of (TOP-REAL 2)

for b, r being Real st A,B,C is_a_triangle & A in circle (a,b,r) & B in circle (a,b,r) & C in circle (a,b,r) & A,B,D is_a_triangle & A in circle (a,b,r) & B in circle (a,b,r) & D in circle (a,b,r) & C <> D & not the_diameter_of_the_circumcircle (A,B,C) = the_diameter_of_the_circumcircle (D,B,C) holds

the_diameter_of_the_circumcircle (A,B,C) = - (the_diameter_of_the_circumcircle (D,B,C))