This paper introduces a new geometric theorem focused on isosceles triangles, specifically examining the properties of perpendicular bisectors. The theorem asserts that in an isosceles triangle, where two sides are equal, the perpendicular bisectors of these sides intersect at a point such that the sum of the lengths of the segments on one bisector is equal to the sum of the lengths of the segments on the other bisector. The theorem is validated through a quantitative research methodology involving geometric constructions, where perpendicular bisectors are drawn for selected isosceles triangles. Mathematical calculations are employed to measure the lengths of the segments formed by the intersection of the bisectors. These segment lengths are then compared and analyzed across various configurations of isosceles triangles, including triangles with different orientations and base lengths. Case studies are conducted to test the theorem under different geometric conditions, ensuring the results are consistent. The methodology is further supported by mathematical proofs, which are derived to formally validate the relationship between the segment sums on each perpendicular bisector. This discovery provides a novel insight into the symmetry of isosceles triangles and contributes to the broader understanding of geometric properties in symmetrical shapes.
S M Nazmuz Sakib (Thu,) studied this question.