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Can this angle triplication construction be considered as a proof of Morley theorem?

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Asked<br>9 months ago

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I am working on a straightedge-and-compass construction that produces the Morley triangle. The method is based on triplicating angles inscribed in a circle, rather than starting from given angle trisectors inside an initial triangle.

Unlike Conway’s construction, where the angles are determined by the initial triangle, here the triplicating angle is inscribed in a circle. This allows it to vary continuously while still generating the Morley configuration for all possible choices of angle and triangle shape.

Construction steps

GeoGebra interactive version

Preparation

Let $A_0A_1$ be a segment of unit length with midpoint $A_{0.5}$.

Draw a circle $u_1$ with radius $A_0A_{0.5}$.

Draw line $A_0A_{0.5}$.

Place a slidable point $A_x$ on $A_0A_{0.5}$ at coordinate $x$.

Mark $A_m$ as the midpoint of $A_0A_x$.

Draw a circle $c_t$ with radius $A_mA_0$.

Place a slidable point $T_A$ on circle $c_t$.

Step sequence for the first vertex

Mark intersections $A_t$ and $A_{-t}$ of $c_t$ with $u_1$.

Draw circle $u_2$ with radius $A_tA_0$. Let $A_{2t}$ be its second intersection with $c_t$.

The chords $A_{-t}A_0$, $A_0A_t$, and $A_tA_{2t}$ are all of equal length $0.5$.

For any position of $T_A$ on arc $A_{2t}A_{-t}$, the lines $T_AA_0$ and $T_AA_t$ are trisectors of angle $\angle A_{-t}T_AA_{2t}$.

By varying $X_A$ and $T_A$, we can construct any angle and its trisectors.

Central vesica piscis circles

Circles $u_1$ and $u_2$ form a vesica piscis.

Draw the outward equilateral circle on chord $A_0A_t$. Its third vertex $A_v$ is the intersection of $u_1$ and $u_2$.

Draw circle $u_3$ with radius $A_vA_0$.

Step sequence for the second vertex

Mark $A_{-v}$ as the second intersection of $u_1$ with $T_AA_{-t}$.

Draw circle $c_v$ through the three points $A_{-v}, A_0, A_v$.

Let $V_A$ be the second intersection of $T_AA_{-v}$ with $c_v$.

Let $A_{2v}$ be the second intersection of $c_v$ with $u_3$.

The chords $A_{-v}A_0$, $A_0A_v$, and $A_vA_{2v}$ are all of equal length $0.5$.

Therefore, lines $V_AA_0$ and $V_AA_v$ trisect angle $\angle A_{2v}V_AA_{-v}$.

Step sequence for the third vertex

Mark $A_{-w}$ as the second intersection of $u_2$ with $T_AA_{2t}$.

Draw circle $c_w$ through the 3 points $A_v, A_t, A_{-w}$.

Let $W_A$ be the second intersection of $T_AA_{-w}$ with $c_w$.

Let $A_{2w}$ be the second intersection of $c_w$ with $u_3$.

The chords $A_{-w}A_t$, $A_tA_v$, and $A_vA_{2w}$ are all of equal length $0.5$.

Therefore, lines $W_AA_v$ and $W_AA_t$ trisect angle $\angle A_{-w}W_AA_{2w}$.

Conclusion

The equilateral triangle $A_0A_tA_v$ is the Morley triangle of $T_AV_AW_A$.

Questions

Can such a triple-angle construction be considered a valid geometric proof of Morley’s theorem? If so, how could it be formalized?

Are there known references to similar “triplication-based” constructions?

geometry<br>triangles<br>circles<br>angle<br>geometric-construction

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edited Sep 27, 2025 at 16:43

asked Aug 10, 2025 at 21:26

Arjen Dijksman

57811 silver badge1212 bronze badges

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$\begingroup$<br>Step 6 is not a valid. How can you make sure that the intersections of $OK, ON$ and the two red circles lie on the given circle (the circle in the 1st step)?<br>$\endgroup$

Duong Ngo

Duong Ngo

2025-09-19 07:20:12 +00:00

Commented<br>Sep 19, 2025 at 7:20

$\begingroup$<br>The intersections of OK, ON and the two red circles don't lie on the red circle of the first step, except if 3β or 3𝛾 is a right angle, see also this question about a special Morley triangle. Or did I misundertand question?<br>$\endgroup$

Arjen Dijksman

Arjen Dijksman

2025-09-19 21:44:04 +00:00

Commented<br>Sep 19, 2025 at 21:44

$\begingroup$<br>I mean you should have made Step 6 more concrete instead of "complete the figure with the triplicated angles $\beta$ and $\gamma$".<br>$\endgroup$

Duong Ngo

Duong Ngo

2025-09-20 05:24:34 +00:00

Commented<br>Sep 20, 2025 at 5:24

$\begingroup$<br>Thanks for your remark. I reworked it and detailed each step.<br>$\endgroup$

Arjen Dijksman

Arjen Dijksman

2025-09-27 16:44:31 +00:00

Commented<br>Sep 27, 2025 at 16:44

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If I'm not mistaken, I think your proof is not complete.

From the construction steps, we can get the following :

Then, you concluded that the equilateral triangle $A_0A_tA_v$ is the Morley triangle of $T_AV_AW_A$.

However, I think in order to get the conclusion, we have to prove that the four points...