An angle trisection

Construction attributed to A. G. O
From page 133 of
Underwood Dudley, The Trisectors, 2nd edition, 1996.

Drag the point $B$ to change the angle $AOB$.
Press “r” to reset the diagram to its initial state.
The red line, $OT$, is an approximate trisector of the angle $AOB$.

Construction

We wish to trisect the given angle $AOB$ represented by the circular arc $AB$ centered at $O$, as shown in the diagram above.

  1. Draw the bisector $OC$ of the angle $AOB$.
  2. Draw the circular arc $DE$ centered at $O$ so that $OD = \frac{1}{3} OA$. Let $G$ be where the line $OC$ intersects the arc $DE$.
  3. Locate $F$ on the extension of $OA$ so that $OF=OD$.
  4. Connect $FG$ and extend to the intersection point $T$ with the arc $AB$.
The line $OT$ (shown in red) is an approximate trisector of the angle $AOB$.

Error Analysis

Let $\alpha$ and $\beta$ be the sizes of the angles $AOB$ and $AOT$, respectively. It is straightforward to show that \[ \beta = \frac{\alpha}{4} + \arcsin\Big( \frac{1}{3}\sin\frac{1}{4}\alpha \Big) = \frac{1}{3}\alpha - \frac{1}{2^4\cdot3^4} \alpha^3 + O(\alpha^7) = \frac{1}{3}\alpha - \frac{1}{1296} \alpha^3 + O(\alpha^7). \] The term after $\alpha^3$ is $\alpha^7$. That's not a typo.

The error $ \ds e(\alpha) = \frac{\alpha}{3} - \beta $ is monotonically increasing in $\alpha$. The worst error on the interval $0 \le \alpha \le \pi/2$ is $e(\pi/2) =$ 0.003 radians = 0.171 degrees. The worst error on the interval $0 \le \alpha \le \pi$ is $e(\pi)$ = 0.024 radians = 1.367 degrees.

This applet was created by Rouben Rostamian using David Joyce's Geometry Applet on July 26, 2002.
Cosmetic revisions on June 13, 2010.

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