Morley's Miracle
H. D. Grossman's Proof
Theorem
| |
The three points of intersection of the adjacent trisectors of the angles of any triangle form an equilateral triangle.
|
Proof
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 ∠CDE = 60° + β and F by making ∠BDF = 60° + γ. Then
| |
∠EDF = 360° - (180° - β - γ) - (60° + β) - (60° + γ) = 60°.
|
Also
| |
∠BFD = 180° - (60° + β + γ) = 60° + α ∠CED.
|
Since D is equidistant from BF and CE, DF = DE and ΔDEF is equilateral.
| |
∠1 = (60° + α) - (β - γ) = 60° - β.
|
Similarly,
Through F draw line r making ∠1' = ang'1. Through E draw a line s making angle ∠2' = ∠2.
| |
∠3 = (60° + α) - (60° - β) = α + β
|
and
Similarly,
Further,
| |
∠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.
- J.Conway's proof
- D. J. Newman's proof
- Bankoff's proof
- B. Bollobás' proof
- Another proof
- Nikos Dergiades' proof
- G. Zsolt Kiss' proof
- M. T. Naraniengar's proof
- Doodling and Miracles
- Morley's Pursuit of Incidence
- Lines, Circles and Beyond
- On Motivation and Understanding
- Bankoff's conundrum
- Of Looking and Seeing
- Morley's Redux and More, Alain Connes' proof
- An Unexpected Variant
- Proof by B. Stonebridge and B. Millar
- Proof by B. Stonebridge
- Proof by Nolan L Aljaddou
- Proof by Roger Smyth
- Proof by H. D. Grossman
- Proof by R. J. Webster
|Contact|
|Front page|
|Contents|
|Geometry|
|Store|
Copyright © 1996-2010 Alexander Bogomolny
|
|
Bookmark
|
Contact
|
Recommend
