Rational Trigonometry: Spread in a circle Part 6
putting everything together: three spreads in a single construction.
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THE figure above has a pair of lines through B (broken) at position ‘1’ on the scale (x-axis) and the respective blue, red and green pependicular lines through A at ‘0’, following the now familiar construction, give the measured spreads OB OR and OG between the lines, depending on the choice of perpendicularity.
In the figure below, however, the same construction is combined with a different (but equivalent) approach: since the (fixed) broken lines are, dualy, perpendicular to each pair of coloured perpendicular lines, the spreads of these lines is measured by the fixed lines providing alternate parallel directions – giving 1 minus the previous values: O’B O’R and O’G in each case. Geometrically this corresponds to reflecting the orginal values in the line x = ½ (shown).
However, a subtly was involved: two pairs of mutually perpendicular lines, just as opposite angles in a cyclic quadrilateral, are supplementary in combination and represent an equal spread. The apparent complementary values of the alternate spreads is due to the axis reversal s‘ = 1 – s, equivalent to invoking perpendicularity.