**F**rom point A draw a vertical line and another line
angling off to the right by angle β which you wish to trisect.
At an arbitrary point B on the vertical line construct a right angle and draw the extended horizontal line BC.
The distance *h* is the length of the hypoteneuse of the triangle BAC, namely AC.

Draw a large number of rays from A, marking their upper endpoints D such that the distance above the extended
horizontal line BC is in each case *2h*.
The loci of the upper endpoints describe a curve known as the Conchoid of Nicomedes ( circa 200 BC).

In polar coordinates, the Conchoid of Nicomedes has *r = a + b*secθ.* See below :-

Forgive the inaccuracy of my hand-sketched (=not constructed) conchoid, please.

**N**ow we
have the conchoid, we can proceed with the actual trisection.
I refer you to my sketch on the left.

From the end of the hypoteneuse at C raise a right angle vertical line from the horizontal side BC. Label the point where it intersects the conchoid D. Actually, this is the only tiny area where we need to construct a segment of the conchoid, we don't need the rest of it :-)

Draw a line DA connecting D on the conchoid with the origin at A. It intersects the horizontal BC at point E. DE=2h, remember.

Now angle BAE is one third of the angle BAC __which we have successfully trisected__ :-)

The "impossible" trick? Note that I subtly deviated from Euclid's *compass and straightedge only* requirement.
When I wrote , I implicitly changed the straightedge into a ruler,
sneaking my change past most people :(

Although my construct shown above does in fact trisect the
angle β (I won't bore you with the proof, suffice that it exists), it does remain __impossible__ to do the trisection
while complying with Euclid's *compass and straightedge only* requirement.

**Comments (3) : **

**Siglinde** (Cheb) complains *"Obviously it only works for acute angles, not for obtuse
or even right angles, so it is at best a partial solution :-("* Mea Culpa.

**Ivan** (Moscow) says *"You also would have to interpolate the conchoid between constructed points, which is
cheating on Euclid too!"*. Yup, you caught that one too.
**Barbara** (15, Florida) noted gleefully *" Even my math teacher didn't know that! I practised at
home with 60° for an hour then showed her today and got a result
measured with a protractor of 20° ! Great Show-and-Tell! Thankyou, Stu :-)"*

Now go visit my blog please, or look at other interesting maths stuff :-)

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