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I hope that the sides opposite to the angles of an equiangular triangle are called homologous sides because these sides are similar.

I would like to know why angles opposite to the sides of an equiangular triangle influence the similarity of these sides. This question developed from this discussion.

Bernard
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justin
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2 Answers2

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If the triangle is equiangular, it is automatically equilateral. Just stop and think; you have three equal angles and you try to form a triangle with them. You cannot stretch nor elongate the sides,otherwise the angles are going to change ; it is perfectly symmetrical.

abiessu
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Elizabeth Middleford
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  • I'm asking about a case where there are two equiangular triangles. Even though they might be equilateral, they might not be mutually equilateral. In other words, they might be similar if not isometric. – justin Jul 18 '18 at 21:47
  • Well, if you want to know why in similar triangles the ratio of any two sides between them would be in proportion, it is because given two similar equilateral triangles, the ratio of either side of the first and either side of the second is the ratio of similarity, given the fact that the sides are the same for each of the figures. The case of isometric triangles is just one particular case. By the way, the geometric transformations that preserve angles of this form are reflection, rotation, translation and homotetia. – Elizabeth Middleford Jul 18 '18 at 22:33
  • I'm looking for an answer regarding the position of angles in two equiangular triangles with it's sides. How does the opposite angles to a given side in these triangles influence the similarity in these equiangular triangles. To be honest I'm not looking for isometric triangles even though isometric triangles are a case of equiangular triangles as you said. – justin Jul 23 '18 at 21:20
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If two triangles have the same corner angles, then those are moved, rotated and/or scaled copies.

--- rk

Dr. Richard Klitzing
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