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T=(\sqrt{T_1}+\sqrt{T_2}+\sqrt{T_3})^2, areas of triangles

Source: 1996 Swedish Mathematical Competition p1

April 2, 2021
triangle areaareageometry

Problem Statement

Through an arbitrary point inside a triangle, lines parallel to the sides of the triangle are drawn, dividing the triangle into three triangles with areas T1,T2,T3T_1,T_2,T_3 and three parallelograms. If TT is the area of the original triangle, prove that T=(T1+T2+T3)2T=(\sqrt{T_1}+\sqrt{T_2}+\sqrt{T_3})^2 .
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