PROBLEM 4. Let $A B C$ be a right-angled triangle at $A$ and $[B D,[C E$ its angle bisectors $(D \in A C$, $E \in A B)$. Denote by $I$ the intersection of lines $B D$ and $C E$ and by $F$, respectively $G$, the projections of points $D$ and $E$ onto line $B C$. Determine the measure of angle $F I G$.[^0]

## NATIONAL MATHEMATICS OLYMPIAD Local stage - 14.02. 2015 GRADING KEY - Grade 7
## PROBLEM 4.

![](https://cdn.mathpix.com/cropped/2024_06_07_c79a5d529e01a215ac3ag-3.jpg?height=434&width=803&top_left_y=1551&top_left_x=638)

(1p) Since $[B I$ and $[B I$ are angle bisectors in $\triangle A B C \Rightarrow [A I$ is an angle bisector in $\triangle A B C$ $\Rightarrow \mathrm{m}(\widehat{B A I})=45^{\circ}$

(2p) Since $[B D$ is the angle bisector of $\widehat{A B C}$, it follows that $D A = D F$ and $A B = B F$ $\triangle A I B \equiv \triangle F I B \Rightarrow \widehat{B A I} \equiv \widehat{B F I} \Rightarrow \mathrm{m}(\widehat{B F I})=45^{\circ}$

(2p) Similarly, it can be shown that $\mathrm{m}(\widehat{F G I})=45^{\circ}$ and from $\Delta F I G$ we have $\mathrm{m}(\widehat{F I G})=90^{\circ}$[^1]


[^0]:    ${ }^{1}$ The actual working time is 3 hours;

    ${ }^{2}$ All problems are mandatory;

    ${ }^{3}$ Each problem is scored from 0 to 7.

[^1]:    ${ }^{1}$ Each corrector awards an integer number of points;

    ${ }^{2}$ Any other correct solution is scored accordingly.