9. Let $A B C$ be a triangle and let $D$ be a point on side $B C$. Suppose that $\widehat{B A D}=\widehat{A C D}$ and $\widehat{C A D}=\widehat{A B D}$. Which of the following statements is certainly true?

(A) $A B C$ is an equilateral triangle

(B) $A B C$ is an obtuse triangle

(C) $A B C$ is an isosceles triangle

(D) $A B C$ is a scalene triangle

(E) $A B C$ is a right triangle

![](https://cdn.mathpix.com/cropped/2024_04_17_fd2d0ad36e2f3d54c0ceg-1.jpg?height=205&width=366&top_left_y=1184&top_left_x=2379)
(9) The correct answer is $(E)$.

Since the angle $\widehat{B A C}$ is equal to the sum of the other two angles of $A B C$, it follows that $\widehat{B A C}$ is a right angle. It is also easily seen that none of the possibilities indicated in the other answers is necessarily true.

Problem proposed by Lorenzo Furio.