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In Plane and solid geometry by Fletcher Durell, he mentions in page 34 that

91. Homologous angles of two mutually equiangular triangles are corresponding angles in those triangles.

Homologous sides of two mutually equiangular triangles are sides opposite homologous angles in those triangles.

I hope to see that, length of an opposite side of the given angle doesn't have to be influenced by it's opposite angle. So why does the author say that "homologous sides are sides opposite to homologous angles"?

I'm asking this question because in some proofs like in Proposition XVI : it says triangle $F'BH$ = triangle $BHC$, so $F'H$ = $CH$ (homologous sides of equal triangles). What is the author trying to say by this statement?

justin
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1 Answers1

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It is a definition, a name to refer to these sides. It does not imply that their length are pairwise equal in general (although the ratio of the lengths is the same for all three pairs).

In the example, it is important to note that it is not "homologous sides of mutually equiangular triangles" but "homologous sides of equal triangles". Equal triangles, in this context, seems to mean isometric.

Arnaud Mortier
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  • Can the term 'corresponding' be used in lieu of 'homologous' or whether 'homologous' has some special meaning other than 'corresponding'? – justin Jul 17 '18 at 21:19
  • @justin Of course you could use that instead, it would also make sense. But the authors of the book want to make everything clear and state which term they are going to use, once and for all. – Arnaud Mortier Jul 17 '18 at 21:26
  • Is there any specific reason, in choosing the side opposite of the homologous angle to be the homologous side of an equianglular triangle? – justin Jul 17 '18 at 21:32
  • @justin If you look at two such triangles, it' clear: homologous sides are those that you would naturally put into pairs if you had to form a correspondence between the sides of the two triangles. The ratio of the lengths is the same for all pairs. – Arnaud Mortier Jul 17 '18 at 21:38
  • Does this mean that for equiangluar triangles, the ratio of the corresponding sides would always be in a proportion? – justin Jul 17 '18 at 21:41
  • @justin Sure, equiangular is a synonym for similar. – Arnaud Mortier Jul 17 '18 at 21:43
  • Could you give a brief proof for the statement that "For equiangular triangles, ratio of corresponding sides would be in a proportion". – justin Jul 17 '18 at 21:48
  • @justin This would need to be a different question. – Arnaud Mortier Jul 17 '18 at 21:49
  • Why is that "homologous sides are those that you would naturally put into pairs if you had to form a correspondence between the sides of the two triangles". Could you help me on this. – justin Jul 17 '18 at 21:58
  • I'm asking: how angles opposite to the sides of an equiangluar triangle influence the similarity of these sides? – justin Jul 17 '18 at 22:07
  • @justin I get you, I mean a different Question altogether on this site - for research and classification purposes. – Arnaud Mortier Jul 17 '18 at 22:11