Consider the following diagram. in $\triangle$ABC:


$\triangle$AOM = a

$\triangle$POC = b

$\triangle$NOC = c

$\triangle$BON = d.

Find the area of $\triangle$MOB and $\triangle$AOP in terms of a,b,c,d. Note that AN, BP and CM are not necessarily medians and $\triangle$ABC is not a special triangle.




Consider the triangle $BOA$ and $OPA$, they share the same height, hence the ratio of their areas is the same as the ratio of their bases: $$\frac{a+y}{x}=\frac{BO}{OP}.$$ The same can be said of the triangles $BOC$ and $POC$: $$\frac{d+c}{b}=\frac{BO}{OP}.$$ By the same argument $$\frac{x+b}{a}=\frac{OC}{MO}=\frac{c+d}{y}.$$ It remains only to solve a system of two equations.

To help to solve the equation we can use the fact that $$adb=xyc$$ which you can check writing each area piece with the formula $1/2l_1l_2 \sin u.$

  • $\begingroup$ I want the solving of the equations. I, myself, got this equations but solving them didn't get into something that makes sense. $\endgroup$ – titansarus May 15 '16 at 15:01
  • $\begingroup$ @titansarus: When you post a problem, you should explain how far you had gotten into solving it and indicate where you got stuck. This helps answerers avoid telling you stuff you already know, and helps them address your specific issues. $\endgroup$ – Blue May 15 '16 at 15:05
  • $\begingroup$ @Blue: Sorry, but I thought my solution is wrong and so i didn't explain that. $\endgroup$ – titansarus May 15 '16 at 15:10
  • $\begingroup$ @Blue excuse me sir. someone told me that it's not appreciated here to edit a post such that we just correct some latex or make the body more clear. is this true ? $\endgroup$ – Arman Malekzadeh May 15 '16 at 15:30
  • $\begingroup$ @titansarus: I understand that you might not want to make your mistakes public. Still, any context is better than no context; just describing your general thoughts about your approach can be helpful. (BTW: You're likely to have your questions put "on hold" if you don't start providing some of these thoughts. We're here to help you understand math, not just to do your homework for you. Giving more than a bare problem statement indicates active participation in the learning process. Please keep this in mind.) $\endgroup$ – Blue May 15 '16 at 15:37

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