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Triangles meeting triangles

Published online by Cambridge University Press:  25 August 2015

Tony Crilly  and
Colin R. Fletcher
Tony Crilly
Affiliation:
10 Lemsford Road, St Albans AL1 3PB e-mail: [email protected]
Colin R. Fletcher
Affiliation:
Atalaya, Lon Glanfred, Llandre, Aberystwyth SY24 5BY e-mail: [email protected]
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Extract

We consider two connected problems:

  • • For a given but otherwise arbitrary triangle in the plane, to construct similar triangles which ‘meet’ this triangle.

  • • To find the triangle so formed which has least area.

1. Constructing a triangle which meets another

These problems beg the question of what is meant by ‘meet’ and we now aim to make this precise:

Definition: A triangle XYZ will meet a given triangle ABC if on the triangle ABC, the vertex X lies on a line through AB, the vertex Y lies on a line through BC, and the vertex Z lies on a line through CA.

When triangle XYZ is actually ‘in’ the triangle ABC, ‘meet’ is synonymous with the traditional ‘inscribe’ (such as in case (1) below). For ‘inscribe’ we understand that some of X, Y, Z may coincide with the vertices of ABC (such as case (2) below).

More generally we use ‘meet’ to extend these possibilities by allowing XYZ to meet triangle ABC with its sides produced externally (such as case (3) below).

Type
Research Article
Information
The Mathematical Gazette , Volume 98 , Issue 543 , November 2014 , pp. 432 - 451
Copyright
Copyright © The Mathematical Association 2014

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References

1. Barry, P., Geometry with trigonometry, Horwood Publishing (2001).Google Scholar
2. Dickson, L.E., History of the theory of numbers, Chelsea (1971).Google Scholar
3. Grattan-Guinness, I., On proving certain optimisation theorems in plane geometry, Math. Gaz. 97 (March 2013) pp. 75–80.Google Scholar