Twin Segments in Arbelos
Here is a problem (10895) from the 2003 American Mathematical Monthly. It was proposed by August Wendijk, Jersey City, NJ. (The solution below is by Thomas Hermann, Milford, OH):
Given a point C on the segment AB, erect semicircles on diameters AB, AC, and BC, all on the same side of AB. Let L be the line through C perpendicular to AB. Let S be the largest circle that fits into the region bounded by L and the semicircles on diameters AC and BC. Let D be the point of tangency between S and the semicircle on BC. Extend the diameter of S through D until it hits L at E. Prove that AC and DE have the same length. |
What if applet does not run? |
References
- August Wendijk; Thomas Hermann, American Mathematical Monthly, Vol. 110, No. 1. (Jan., 2003), pp. 63-64.
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Copyright © 1996-2018 Alexander BogomolnyLet's introduce the following notations
- a the radius of the circle on AC as a diameter,
- G the center of the circle with diameter BC; b the radius of the circle,
- H the center of the circle with diameter AB; R the radius of the circle,
- F the center of the Archimedian twin in the problem (the largest circle that fits into the region bounded by L and the semicircles on diameters AC and BC); r - the radius of the the circle,
- M the projection of F on AB; y = FM,
- T the point of tangency of circles H(R) and F(r),
- w = DE.
We have to show that w = 2a.
What if applet does not run? |
Observe that by the construction,
- CM = r,
- HM = |b - a - r|,
- GF = b + r,
- FH = a + b - r.
Triangles FGM and FHM are right-angled. The Pythagorean theorem implies:
(b + r)² = y² + (b - r)² and (a + b - r)² = y² + (b - a - r)², |
from which
r = ab / (a + b). |
Triangles FGM and CEG are similar so that
(b + r) / (b - r) = (b + w) / b. |
From the latter
r = bw / (2b + w). |
Comparing the to expressions for r yields w = 2a, as needed.
- Arbelos - the Shoemaker's Knife
- 7 = 2 + 5 Sangaku
- Another Pair of Twins in Arbelos
- Archimedes' Quadruplets
- Archimedes' Twin Circles and a Brother
- Book of Lemmas: Proposition 5
- Book of Lemmas: Proposition 6
- Chain of Inscribed Circles
- Concurrency in Arbelos
- Concyclic Points in Arbelos
- Ellipse in Arbelos
- Gothic Arc
- Pappus Sangaku
- Rectangle in Arbelos
- Squares in Arbelos
- The Area of Arbelos
- Twin Segments in Arbelos
- Two Arbelos, Two Chains
- A Newly Born Pair of Siblings to Archimedes' Twins
- Concurrence in Arbelos
- Arbelos' Morsels
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