area chasing inside a rectangle (2009 Belgium MO OMB p1)

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High School Math Grades 9-12, Ages 13-18

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area chasing inside a rectangle (2009 Belgium MO OMB p1)

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#1 h Aug 10, 2020, 10:43 AM

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Let $M$ and $N$ , respectively, be the points of the sides $[AB]$ and $[BC]$ of a rectangle $ABCD$ . Let $a,b,c$ be the area of the triangle $AMD,MBN, NCD$ respectively. Express in terms of $a,b$ and $c$ , the area of the triangle $DMN$ .

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#2 h Aug 11, 2020, 3:36 PM • 1 Y

Y by Mango247

https://lh4.googleusercontent.com/Io39c9cGUb5a--wMpiYNnzU_xMIJUN5CGpSlce4ireLDrT8vTU4rNwK7gZ8IWBs1-W3Y_dux6GtUau0G2b8L3a1XxOyJovQh63QlcIw5Fg7hxpF6R0WyJc1StW7W8hbcz0CBSYqS

some parts

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#3 h Aug 11, 2020, 4:06 PM • 4 Y

Y by jL56L06B9, Mango247, Mango247, Mango247

Let $AD=x$ , $CD=y$ . Then we have $AM=2a/x$ , $CN=2c/y$ , $2b=(y-2a/x)(x-2c/y)$ . Expanding we get a quadratic in $xy$ , and we find $xy=a+b+c+\sqrt{(a+b+c)^2-4ac}$ . So our desired area is $xy-a-b-c$ , or $\sqrt{(a+b+c)^2-4ac}$ .

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