# Part 6

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#### Rational Trigonometry:** Spread in a circle**** ****Part 6**

**Spread in a circle**

**– Part 1 – Part 2 – Part 3 – Part 4 – Part 5 –**

### putting everything together: three spreads in a single construction.

Click for a larger image

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 O_{B} O_{R} and O_{G} 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.

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