Exercise 9. Let $A M C$ be an isosceles triangle with $M$ as the vertex and such that the angle $\widehat{A M C}$ is acute. Let $B$ be the symmetric point of $A$ with respect to $M$ and let $H$ be the foot of the altitude from vertex $C$ in the triangle $A B C$. Suppose that $A H = H M$. Calculate the values of the three angles of the triangle $A B C$.
## Solution to Exercise 9

![](https://cdn.mathpix.com/cropped/2024_06_04_01f3cce223794094ee24g-16.jpg?height=508&width=759&top_left_y=403&top_left_x=660)

In triangle $A M C$, the line $(C H)$ is the height and the median from vertex $C$. Therefore, triangle $A C M$ is isosceles at $C$. Hence, $A C = C M$. Since $C M = M A$, triangle $A C M$ is equilateral and $\widehat{B A C} = 60^{\circ}$.

Since points $A, M$, and $B$ are collinear, we have $\widehat{C M B} = 180^{\circ} - \widehat{A M C} = 120^{\circ}$.

Since $A$ and $B$ are symmetric with respect to $M$, we have $M A = M B = M C$, so triangle $M B C$ is isosceles at $M$. Thus,

$$
180^{\circ} = \widehat{B M C} + \widehat{M B C} + \widehat{B C M} = \widehat{B M C} + 2 \widehat{M B C}
$$

Therefore, $2 \widehat{M B C} = 180^{\circ} - \widehat{B M C} = 60^{\circ}$ and $\widehat{A B C} = 30^{\circ}$.

We finally deduce that $\widehat{A C B} = \widehat{A C M} + \widehat{M C B} = 60^{\circ} + 30^{\circ} = 90^{\circ}$.

The angles of the triangle are therefore $(\widehat{A B C}, \widehat{B C A}, \widehat{C A B}) = \left(30^{\circ}, 90^{\circ}, 60^{\circ}\right)$.

Grader's Comment: The exercise was generally very well done. Students were able to use the hypotheses in the problem statement to determine the angles of the triangle, and several methods were proposed.

Most of the time, students who did not solve the exercise did not exploit the equality $H M = H A$ given in the problem statement. It is crucial to try to use all the hypotheses in the problem statement: any reasoning that does not use them in full was bound to fail here. Checking that all hypotheses have been invoked is also a good way to review one's reasoning.

Other students lost points in their justifications. Be sure to clearly justify your arguments, especially to prove that triangle $A M C$ is equilateral: observing a property on a large number of figures is not enough to prove it.

Moreover, trigonometric or analytical approaches, which are generally risky, often resulted in a loss of points due to calculation errors.

Finally, it is unfortunate that some students lose points unnecessarily due to a poor reading of the problem statement. We also remind students of the importance of making a clear diagram to avoid this.