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Thread: triangle question

  1. #1
    Member disclaimer's Avatar
    Dec 2007

    triangle question

    Hi, here is my problem:

    Inside of a triangle $\displaystyle ABC$, a random point $\displaystyle M$, in which three straight lines parallel to the respective sides of the triangle intersect, was chosen. Those straight lines have divided the triangle into six parts, three of which are also triangles. Let $\displaystyle r_1, r_2, r_3$ be the radii of circles inscribed in those three triangles, and let $\displaystyle r$ be the radius of a circle inscribed in triangle $\displaystyle ABC$. Show that $\displaystyle r=r_1+r_2+r_3$.

    Help appreciated. Thank you.
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  2. #2
    Moo is offline
    A Cute Angle Moo's Avatar
    Mar 2008
    P(I'm here)=1/3, P(I'm there)=t+1/3
    I found it !

    Okay look at the sketch. It is not necessary to draw the inscribed circles.

    triangle question-inscribed.jpg

    Consider triangles DEM and ABC. All their sides are parallel to each other. Hence they're congruent (or similar). Let $\displaystyle k_1$ be the ratio between the two. This means that $\displaystyle k_1=\frac{DE}{AB}=\frac{DM}{AC}={\color{red}\frac{ EM}{BC}}={\color{blue}\frac{r_1}{r}}=\dots$
    This always holds true for two congruent triangles.

    It'll be the same for triangle FGM : $\displaystyle k_2={\color{blue}\frac{r_2}{r}}={\color{red}\frac{ MG}{BC}}=\dots$

    and for triangle HMI : $\displaystyle k_3={\color{blue}\frac{r_3}{r}}={\color{red}\frac{ HI}{BC}}$

    Now consider the blue parts of the equalities.
    You can see that if you sum them, it should give 1, according to the result you're given.

    So we'll prove that the sum of the red parts is equal to 1 !

    $\displaystyle {\color{red}\frac{EM}{BC}+\frac{MG}{BC}+\frac{HI}{ BC}}$

    But since EM//BH and EB//MH (parallel), then EMHB is a parallelogram. Hence $\displaystyle \bold{EM=BH}$

    Similarly, MGCI is a parallelogram. And hence $\displaystyle \bold{MG=IC}$

    So the red sum becomes :
    $\displaystyle =\frac{BH}{BC}+\frac{IC}{BC}+\frac{HI}{BC}=\frac{B H+IC+HI}{BC}$

    you can see that the points B, H, I and C are aligned. Hence $\displaystyle BH+IC+HI=BC$

    So the red part is equal to 1.

    Therefore for the blue part :
    $\displaystyle 1=\frac{EM}{BC}+\frac{MG}{BC}+\frac{HI}{BC}=\frac{ r_1}{r}+\frac{r_2}{r}+\frac{r_3}{r}=\frac{r_1+r_2+ r_3}{r}$

    And you're done
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