P
roblem
Is the statement “The bisector of the apex angle of a triangle and the perpendicular bisector of the opposite side intersect at a point outside the triangle” correct?
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Solution
The bisector of the apex angle of a triangle and the perpendicular bisector of the opposite side generally pass through the same point $G$ on the circumference of the triangle’s circumcircle. However, in an isosceles triangle, these two lines overlap. Therefore, this statement is incorrect.
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Reference Teiichiro Sasabe (1976) The Encyclopedia of Geometry (2nd edition), Seikyo-Shinsha, p.19