theorem Th47: :: EUCLID12:62

for A, B, C being Point of (TOP-REAL 2) st A,B,C is_a_triangle holds

ex D being Point of (TOP-REAL 2) st

( (the_perpendicular_bisector (A,B)) /\ (the_perpendicular_bisector (B,C)) = {D} & (the_perpendicular_bisector (B,C)) /\ (the_perpendicular_bisector (C,A)) = {D} & (the_perpendicular_bisector (C,A)) /\ (the_perpendicular_bisector (A,B)) = {D} & |.(D - A).| = |.(D - B).| & |.(D - A).| = |.(D - C).| & |.(D - B).| = |.(D - C).| )

ex D being Point of (TOP-REAL 2) st

( (the_perpendicular_bisector (A,B)) /\ (the_perpendicular_bisector (B,C)) = {D} & (the_perpendicular_bisector (B,C)) /\ (the_perpendicular_bisector (C,A)) = {D} & (the_perpendicular_bisector (C,A)) /\ (the_perpendicular_bisector (A,B)) = {D} & |.(D - A).| = |.(D - B).| & |.(D - A).| = |.(D - C).| & |.(D - B).| = |.(D - C).| )