
The figure on the left shows a geometrical demonstration of Euclid's second
theorem (Corollary to Proposition 8 of Book VI): " If in a rightangled triangle a perpendicular is drawn from the right angle to the base, then the straight line so drawn is a mean proportional between the segments of the base "
 
Let the triangle be ABC, with right angle C. AB is the base and CH is the height. The proposition says that This means that the square over CH has the same area as the rectangle with sides AH and HB.
By rotating the triangle HAC 90° clockwise around H we get
the triangle HA"C", and
rotating it counterclockwise we get HA'C'.
Thus C'H and HC are two sides of the square C'HCE.
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