北京市十一学校：On Spiral Similarity In Families of Simson Polygons

P
to the three sides of
Simson Line theorem, possibly one of the most well-known mathematical theorem, establishes the spatial relationship of collinearity between the projections of a point
A33 A32 P = A03 A02 P,
A31 A32 P = A01 A02 P,
A32 A31 P = A02 A01 P,
A32 A33 P =
A02 A03 P
∴
A32 A31 A33 = A32 A31 P + P A31 A33 = A02 A01 P + P A01 A03 = A02 A01 A03
are cyclic quadrilaterals.
A02 A01 P =
A13 A01 P = A13 A12 P
A01 A02 P = A13 A11 P,
A03 A02 P =
A11 A13 P,
A02 A03 P = A11 A12 P,
A01 A03 P = A12 A11 P,
Then, for by
Proposition
point
A13 .
to have
A01 A02 A03 and a point P , project P to line A01 A02 at A12 , A11 similarly. Repeat the projection from P on A11 A12 A13 A21 A22 A23 , and then A31 A32 A33 , then
On Spiral Similarity in Families of Simson Polygons
Authors: Dekun Song, Haozhe Zhang, Zihan Zhang 17 August 2017
Beijing National Day School, 66 Yuquan Rd, Haidian Qu, Beijing, China
2
Notations and Concepts
In some sections of the following discussion, we use di-
Directed Angle
180. In other words, ABC = − CBA = , ABC + 180 . Some important corollaries are ABC + BCA + CAB = 0, P BA = P BC ⇔ A, B , C are collinear, DAB = DCB ⇔A, B , C , D are concyclic. Moreover, ABC is dened to be equal to its numerical value when points A, C are in an anticlockwise order with respect to point B .
A03 A01 P = A12 A13 P
A21 A22 A23 , we can increase the rst number in each subscription
1
and have
4
A12 A11 P = A23 A22 P,
A11 A12 P =
A23 A21 P,
A13 A12 P = A21 A23 P,
ABC (AB , BC , CA)
that is concyclic with
the point of projection. When the four points are not colinear, the projections become a triangle, which some identify as the Simson triangle.
A12 A13 P = A21 A22 P,
A11 A13 P = A22 A21 P,
A13 A11 P =
A22 A23 P
Due to the rst set of equalities, there is also
A23 A22 P = A01 A03 P,
A23 A21 P =
5
A02 A03 P,
For Dene
A31 A32 A33 ∼
Proof:
A01 A02 A03
⇒ P A12 ⊥ A01 A03 , P A11 ⊥ A03 A02 , P A13 ⊥ A02 A01
3
∴ A12 A01 A13 P, A13 A02 A11 P, P A11 A03 A12 ∴
Similarly,
Research Topic: On Spiral Similarity in Families of Simson Polygons Keywords: Simson polygon, Cevian product, spiral similarity, projection
2
1
Introduction to the Research Topic
ABC ,
thus in both conditions the proposition has been proved.
The key to the proof is the observation of the permutation of angles that is delivered by the concylic quadrilaterals. Since the permutation has a cycle of 3, the triangle
A33 A31 P,
A23 A22 P = A31 A33 P,
A22 A23 P = A31 A32 P,
Yet due to equalities
A21 A23 P = A32 A31 P,
A23 A21 P =
A32 A33 P
(∗), A33 A31 P = A03 A01 P, A31 A33 P = A01 A03 P,
Tutor: Yue Guan
1
Abstract
Starting from the concept of Simson triangle, the paper establishes the term Simson polygon, and examines the iteration of generating Simson polygons, namely projecting one point onto the sides of the original polygon. The paper manages to prove that some pairs of Simson polygons have relationships of spiral similarity and oers insights on the maximization of the ratio of similarity between the two polygons, dened in the paper as the Cevian product of the point of projection about the original polygon. The particularity of the research is its comprehensiveness within its eld: Euclidean geometry. Beginning from rather simple constructions of cyclic quadrilaterals, we proved the similarity through a basic anglechasing method that has been backed up by the concept of permutations. Then, we dealed with the ratio of similarity combining both trignometry and algebraic calculations of line segments' ratios. Approaching the maximization of the Cevian product about triangles, we utilized translation, the geometric transformation by applying Pappus' theorem. Thereafter, more algebra has been introduced to tackle more complicated problems: dierentiation was used to maximize the Cevian product of polygons, and locus of points with a xed Cevian product has been examined in a Cartesian way. Virtually all methods in classical Euclidean geometry has been used (one exception is vectors) for the research, and, despite the relatively limited application of our results in the real life, the methods in our research problems have indeed presented the vast beauty of geometry.