Let $ABC$ be a triangle such that $BA > BC$. Let $M$ be the midpoint of $AC$, the internal bisector of angle $\widehat{B}$ intersects $AC$ at $I$. The line parallel to $BC$ passing through $I$ intersects $AM$ at $E$. The line parallel to $BA$ passing through $M$ intersects $AI$ at $D$. What is the measure of angle $\widehat{EDI}$?
![](https://cdn.mathpix.com/cropped/2024_05_10_59677aa68548da326d08g-255.jpg?height=586&width=946&top_left_y=1721&top_left_x=557)

We draw some nice figures, which allow us to conjecture that the angle in question is $90^{\circ}$. We then look at what information we can obtain from our figure. We introduce $N$ (resp. $P$) as the intersection of $EM$ with $BC$ (resp. $DI$). Thales' theorem applied in $ABC$ tells us that $BN = NC$. We reapply it in $BNM$ and $NMC$ to obtain that $DP / BN = MP / MN = IP / NC$, and thus $PD = PI$. Thanks to alternate interior angles, opposite angles, and the definition of the angle bisector, we get: $\widehat{\mathrm{PIE}} = \widehat{\mathrm{IBC}} = \widehat{\mathrm{ABI}} = \widehat{\mathrm{BEN}} = \widehat{\mathrm{PEI}}$ and thus $\mathrm{PEI}$ is isosceles and $\mathrm{PE} = \mathrm{PD} = \mathrm{PI}$. Therefore, $\mathrm{D}$ lies on the circle with diameter $EI$, so the desired angle is $90^{\circ}$.

A remark: the assumption $BA > BC$, given so that $I$ is "to the left of $M$" in the figure, is not necessary; the problem works even if $BC > BA$.

## 3 Geometry Course/TD

See the geometry course of group C, starting from page 60.

## 3 Group $\mathbf{C}$

## 1 Arithmetic Course

- GCD Calculations -

We denote $(a, b)$ as the GCD of integers $a$ and $b$. We recall that $(a, b) = (a, b + ca)$ with $a, b, c$ being integers. This is the basis of the Euclidean algorithm for calculating the GCD of two integers: if $a = bq + r, 0 \leq r < a$ is the Euclidean division of $a$ by $b$, then $(a, b) = (r, b) = (b, r)$, and by calculating the sequence of remainders of the Euclidean division of the first argument by the second, we obtain a strictly decreasing sequence of positive remainders, which eventually reaches 0. The last step of the calculation thus yields a natural number $d$ such that $(a, b) = (d, 0) = d$, and we have thus calculated the GCD of $a$ and $b$.