Let angle BCA = x and angle CAB = y, as shown in the figure above.
Since x+y=90, the result is the following figure:
Every triangle in the figure above has the same combination of angles: x-y-90
Implication:
The triangles are all SIMILAR.
Let BC=4, yielding the following figure:
In triangle ABC, the sides are in the following ratio: 2-4-2√5.
Since the triangles are all similar, each must be composed of three sides in this ratio.
The result is the following figure:
Thus:
Red triangle \(= \frac{1}{2}*1*2 = 1\)
Red rectangle = ACKJ - ACDH \(= (2√5)^2 - (2√5 * \frac{8}{√5}) = 20-16 = 4\)
Total area = ACKJ+ AFG + ABC \(= (2√5)^2 + (\frac{1}{2}*1*2) + (\frac{1}{2}*2*4) = 20+1+4 = 25\)
\(\frac{red}{total} = \frac{1+4}{25} = \frac{5}{25} = \frac{1}{5}\)
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