Legendre’s theorem on angles of triangle
Adrien-Marie Legendre has proved some theorems concerning the sum of the angles of triangle. Here we give one of them, being the inverse of the theorem in the entry “sum of angles of triangle in Euclidean geometry”.
Theorem. If the sum of the interior angles of every triangle equals straight angle, then the parallel postulate is true, i.e., in the plane determined by a line and a point outwards it there is exactly one line through the point which does not intersect the line.
Proof. We consider a line and a point not belonging to . Let be the normal line of (with ) and be the normal line of through the point . By the supposition of the theorem, does not intersect .
We will show that in the plane determined by the line and the point , there are through no other lines than not intersecting the line . For this purpose, we choose through a line which differs from ; let the line form with an acute angle .
We determine on the line a point such that . By the supposition of the theorem, in the isosceles right triangle we have
Next we determine on a second point such that . By the supposition of the theorem, in the isosceles triangle we have
We continue similarly by forming isosceles triangles using the points , , , of the line such that
Then the acute angles being formed beside the points are
Then the line falls after penetrating into the inner territory of the triangle . Thereafter it must leave from there and thus intersect the side of this triangle. Accordingly, intersects the line .
The above reasoning is possible for each line through . Consequently, the parallel axiom is in force.
- 1 Karl Ariva: Lobatsevski geomeetria. Kirjastus “Valgus”, Tallinn (1992).
|Title||Legendre’s theorem on angles of triangle|
|Date of creation||2013-05-11 13:55:53|
|Last modified on||2013-05-11 13:55:53|
|Last modified by||pahio (2872)|