12) In a triangle, for every pair of consecutive sides, the two perpendicular bisectors of the sides and the angle bisector of the angle formed by the two sides intersect at the same point. We can state that:

(A) there is no triangle with this property, (B) the triangle is equilateral, (C) the triangle has an angle of $30^{\circ}$, (D) the triangle is right-angled, (E) the triangle has an angle of $45^{\circ}$.
12. The answer is (B). Referring to the figure, consider the two consecutive sides $C A$ and $A B$. The segments $H O$ and $H^{\prime} O$ belong to the perpendicular bisectors of $A B$ and $C A$ respectively, and $O$ is the point where these perpendicular bisectors intersect the angle bisector of $\widehat{C A B}$. The triangles $A H O$ and $A H^{\prime} O$ both have a right angle and have one equal angle: $\widehat{H^{\prime} A O}=\widehat{H A O}$, so they have all angles equal; moreover, they share the side $A O$ and thus are congruent triangles. Consequently, $A H$ and $A H^{\prime}$ have the same length; therefore, $A B$ and $C A$ also have the same length, since $H$ and $H^{\prime}$ are the midpoints of $A B$ and $C A$ respectively (the perpendicular bisectors intersect the sides at their midpoints). This reasoning can be repeated for the sides $A B$ and $B C$, which are therefore equal. In conclusion, the sides of triangle $A B C$ are all equal to each other, and thus the triangle is equilateral.

![](https://cdn.mathpix.com/cropped/2024_04_17_d9e56cae5925cb0c0feag-4.jpg?height=382&width=437&top_left_y=343&top_left_x=861)