H. D. Grossman's Proof
The three points of intersection of the adjacent trisectors of the angles of any triangle form an equilateral triangle.
This proof was published in The American Mathematical Monthly, Vol. 50, No. 9 (Nov., 1943), p. 552.
Let the triangle have base BC and angles 3α, 3β,3γ. Let BDK, BF, CDH, CE be angle trisectors. E is determined by making
∠EDF = 360° - (180° - β - γ) - (60° + β) - (60° + γ) = 60°.
∠BFD = 180° - (60° + β + γ) = 60° + α.
Similarly, ∠CED = 60° + α.
Since D is equidistant from BF and CE,
∠1 = (60° + α) - (β - γ) = 60° - β.
∠2 = 60° - γ.
Through F draw line r making
∠3 = (60° + α) - (60° - β) = α + β
∠mr = (α + β) - β = α.
∠sn = (α + γ) - γ = α.
∠mn = (180° - 3β - 3γ) = 3α.
It remains only to prove that the lines m, n, r, and s converge to a point. The line KF joins the vertices of two isosceles triangles and therefore bisects ∠K. Then in triangle mBKs the bisector of ∠ms passes through F and being parallel to r, coincides with it. Similarly in triangle rHCn the bisector of ∠rn passes through E and being parallel to s, coincides with it.
On Morley and his theorem
- Doodling and Miracles
- Morley's Pursuit of Incidence
- Lines, Circles and Beyond
- On Motivation and Understanding
- Of Looking and Seeing
- J.Conway's proof
- D. J. Newman's proof
- B. Bollobás' proof
- G. Zsolt Kiss' proof
- Backward Proof by B. Stonebridge
- Morley's Equilaterals, Spiridon A. Kuruklis' proof
- J. Arioni's Proof of Morley's Theorem
- Bankoff's proof
- B. Bollobás' trigonometric proof
- Proof by R. J. Webster
- A Vector-based Proof of Morley's Trisector Theorem
- L. Giugiuc's Proof of Morley's Theorem
- Dijkstra's Proof of Morley's Theorem
- Another proof
- Nikos Dergiades' proof
- M. T. Naraniengar's proof
- An Unexpected Variant
- Proof by B. Stonebridge and B. Millar
- Proof by B. Stonebridge
- Proof by Roger Smyth
- Proof by H. D. Grossman
- Proof by H. Shutrick
- Original Taylor and Marr's Proof of Morley's Theorem
- Taylor and Marr's Proof - R. A. Johnson's Version
- Morley's Theorem: Second Proof by Roger Smyth
- Proof by A. Robson
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