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Tweedie's Theorem

Consider two directly similar triangles DeltaA_1B_1C_1 and DeltaA_2B_2C_2 with

B_1C_1:A_1C_1:A_1B_1=B_2C_2:A_2C_2:A_2B_2=a:b:c.

Then a·A_1A_2, b·B_1B_2 and c·C_1C_2 form the sides of a triangle. The triangle is degenerate if and only if the similitude center of DeltaA_1B_1C_1 and DeltaA_2B_2C_2 lies on the circumcircle of these triangles.

This theorem extends Ptolemy's theorem and the Ptolemy inequality.

See also

Pompeiu's Theorem, Ptolemy Inequality, Ptolemy's Theorem

This entry contributed by Floor van Lamoen

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References

Bottema, O. Verscheidenheden. Groningen, Netherlands: Wolters-Noordhoff/NVvW, pp. 134-137, 1977.Pinkerton, P. "Note on Mr. Tweedie's Theorem in Geometry." Edinburgh Math. Soc. Proc. 22, 27, 1904.Tweedie, C. "Inequality Theorem Regarding the Lines Joining Corresponding Vertices of Two Equilateral, or Directly Similar, Triangles." Edinburgh Math. Soc. Proc. 22, 22-26, 1904.

Referenced on Wolfram|Alpha

Tweedie's Theorem

Cite this as:

van Lamoen, Floor. "Tweedie's Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/TweediesTheorem.html

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