In the figure below, area of triangles $AOD, DOC,$ and $AOB$ is given. Find the area of triangle $OEF$ in terms of area of these three triangles. [asy][asy] import graph; size(11.52cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-2.4,xmax=9.12,ymin=-6.6,ymax=5.16; pair A=(0,0), F=(9,0), B=(4,0), C=(3.5,2), D=(1.94,2.59), O=(2.75,1.57); draw(A--(3,4),linewidth(1.2)); draw((3,4)--F,linewidth(1.2)); draw(A--F,linewidth(1.2)); draw((3,4)--B,linewidth(1.2)); draw(A--C,linewidth(1.2)); draw(B--D,linewidth(1.2)); draw((3,4)--O,linewidth(1.2)); draw(C--F,linewidth(1.2)); draw(F--O,linewidth(1.2)); dot(A,ds); label("$A$",(-0.28,-0.23),NE*lsf); dot(F,ds); label("$F$",(8.79,-0.4),NE*lsf); dot((3,4),ds); label("$E$",(3.05,4.08),NE*lsf); dot(B,ds); label("$B$",(4.05,0.09),NE*lsf); dot(C,ds); label("$C$",(3.55,2.08),NE*lsf); dot(D,ds); label("$D$",(1.76,2.71),NE*lsf); dot(O,ds); label("$O$",(2.57,1.17),NE*lsf); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy][/asy]
Problem
Source: Iran Third Round Problems 1993 – Poblem 2
Tags: geometry, geometry proposed
29.07.2011 22:38
What property does $F$ have?
24.08.2011 19:53
If a = area of AOD, b = area of DOC, c = area of AOB, the area of OEF is 2bc(a + b)(a + c)/[(c - b)(a^2 - bc)], and this is a correct answer confirmed by software.
24.08.2011 20:44
Vo Duc Dien wrote: If a = area of AOD, b = area of DOC, c = area of AOB, the area of OEF is 2bc(a + b)(a + c)/[(c - b)(a^2 - bc)], and this is a correct answer confirmed by software. Good job software! May we have the solution as the worth of solution is more than the answer
25.08.2011 08:05
Spats wrote: What property does $F$ have? It looks like F, C, D are supposed to be collinear. I'm still kind of confused how all of these points are defined... [asy][asy] import graph; size(11.52cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-2.4,xmax=9.12,ymin=-6.6,ymax=5.16; pair A=(0,0), F=(9,0), B=(4,0), C=(3.5,2), D=(1.94,2.59), O=(2.75,1.57); draw(A--(3,4),linewidth(1.2)); draw((3,4)--F,linewidth(1.2)); draw(A--F,linewidth(1.2)); draw((3,4)--B,linewidth(1.2)); draw(A--C,linewidth(1.2)); draw(B--D,linewidth(1.2)); draw((3,4)--O,linewidth(1.2)); draw(C--F,linewidth(1.2)); draw(F--O,linewidth(1.2)); dot(A,ds); label("$A$",(-0.28,-0.23),NE*lsf); dot(F,ds); label("$F$",(8.79,-0.4),NE*lsf); dot((3,4),ds); label("$E$",(3.05,4.08),NE*lsf); dot(B,ds); label("$B$",(4.05,0.09),NE*lsf); dot(C,ds); label("$C$",(3.55,2.08),NE*lsf); dot(D,ds); label("$D$",(1.76,2.71),NE*lsf); dot(O,ds); label("$O$",(2.57,1.17),NE*lsf); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); draw(C--D, dashed);[/asy][/asy]