Around the Incircle
In a curious twist several geometric constructs emerge (or merge) in a single dynamic diagram:
- H. Eves' problem of getting 6 concyclic points by rotating the side lines of a triangle around its incircle.
- Adams' circle is centered at the incenter and passes through the points of intersection of Gergonne's lines with the side lines of the given triangle.
- Conway's circle passes through the six points obtained by producing the sides of
 ABC a units beyond A, b units beyond B, and c units beyond C.
- A construction of shapes of constant width.
Do you see how?
Explanation
Copyright © 1996-2009 Alexander Bogomolny
Let's consider the problems one at a time.
Howard Eves' Quickie
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Starting at a point P on the side BC of DABC, mark Q on AB with BQ = BP, R on AC with AR = AQ, P' on BC with CP' = CR, Q' on AB with BQ' = BP', and so on. Prove that the construction closes, i.e., that CP = CR', and that the six points P, Q, R, P', Q', R' lie on the same circle.
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Solution
Let D, E, F be the points of tangency of the incircle of DABC with the sides BC, AC, and AB, respectively. Then, say, BD = BF, because both are tangent to the incircle from point B. Therefore, DP = FQ. Subsequently,
| (1) |
FQ = ER = DP' = FQ' = ER' = DP.
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The process indeed is getting closed.
Now, points D, E, F are equidistant from the center I of the incircle. The radius-vectors ID, IE, IF are perpendicular to the sides of DABC. Additionally, D, E, F serve as the midpoints of equal segments PP', RR', QQ'. Which implies that the six points P, P', Q, Q', R, R' are equidistant from I.
Note that if DP = 0, to start with, then after three rotations DP' = 0. Which means that point D is fixed by successive rotations around B, A and C.
(It is worth noting that the construction can be expanded to n-gons, with n > 3.)
Adams' Circle
This is a particular case of the above situation where P is chosen so that the lines PR', RQ, and P'Q' are concurrent. Wonderfully, they meet at the Gergonne point of DABC.
Conway's Circle
Conway's circle are obtained by producing the sides AB and AC beyond A by a (= BC), etc. Concentric circles are also obtained if all the extension are modified by a fixed amount, say, x. ("Beyond" should be understood here metaphorically as "away from".)
Proof
We know that
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CE = CD = (a + b - c)/2,
BD = BF = (a - b + c)/2,
AE = EF = (-a + b + c)/2.
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If AC is produced beyond C by (c+x) to, say, C', and beyond A by (a+x) to, say, A', then A'C' = (a + b + c + 2x) and
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| EA' | = (a + x) + (-a + b + c)/2 |
| | = (a + b + c)/2 + x |
| | = (a + b - c)/2 + (c + x) |
| | = EC'. |
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The situation is exactly as before for the points R, E, R'. Conway's construction leads to exactly the circles in Eves' problem.
Curves of Constant Width
Move point P away so that all six points D, E, F, D', E', F' are outside DABC.
You should get something like
Eves' arcs form a convex shape with an interesting property: In whatever direction it is squeezed between two parallel lines the distance between the lines remains the same. Figures with this property are known as shapes (or curves) of constant width. Shapes of constant width can be constructed starting with equilateral but not necessarily equiangular stars. Eves' arcs suggest a different approach that generalizes the crossed-lines method discussed in [Gardner, p. 217, Honsberger, p. 158, Rademacher, p. 167]. The crossed-lines method no longer requires that we start with an equilateral triangle.
Note that three segments PP', QQ', RR' all have the same length (a + b + c)/2 + x, for some x. When any of these is rotated around any of its points, the two arcs thus obtained may only be enclosed (locally, i.e., away from their extremities) between parallel lines at the distance (a + b + c)/2 + x. For the segment will have to be perpendicular to the two lines. When the three pairs of arcs stick together as in the case of Eves' arcs at hand, we obtain exactly the condition that defines the shapes of constant width.
References
- M. Gardner, The Unexpected Hanging and Other Mathematical Diversions, The University of Chicago Press, 1991
- R. Honsberger, Ingenuity in Mathematics, MAA, New Math Library, 1970
- H. Rademacher and O.Toeplitz, The Enjoyment of Mathematics, Dover Publications, 1990.
- C. W. Trigg, Mathematical Quickies, Dover, 1985, #181 (Am Math Monthly, 50 (June, 1943), 391)
Copyright © 1996-2009 Alexander Bogomolny
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