A proof or brief explanation why would be nice too! Thank you!!
Hello, Jammix!
Divide a triangle in two equal areas by the line parallel to its base.The large triangle has base $\displaystyle B$ and height $\displaystyle H.$Code:- * : *: * : * :h * H * : * : * * * * * * * : * b * - * * * * * * * * * * B
The small triangle has base $\displaystyle b$ and height $\displaystyle h.$
We want: .$\displaystyle \tfrac{1}{2}bh \:=\:\tfrac{1}{2}\left(\tfrac{1}{2}BH\right) \quad\Rightarrow\quad bh \:=\:\tfrac{1}{2}BH\;\;[1]$
From similar triangles: .$\displaystyle \frac{b}{h} \,=\,\frac{B}{H} \quad\Rightarrow\quad b \,=\,\tfrac{B}{H}h$
Substitute into [1]: .$\displaystyle \left(\tfrac{B}{H}h\right)h \:=\:\tfrac{1}{2}BH \quad\Rightarrow\quad h^2 \:=\:\tfrac{1}{2}H^2$
Therefore: .$\displaystyle h \:=\:\frac{H}{\sqrt{2}}$