In Euclidean geometry, the median theorem , or Apollonius theorem , designates one of the following three identities [ first ] , on distances and scalar products, in a triangle ABC median WHO and high Like :
First median theorem or Apollonius theorem [ modifier | Modifier and code ]
Apollonius theorem – Be ( ABC ) Any triangle and WHO the median from A . We then have the following relation:
or :
This theorem is a reformulation of the identity of the parallelogram.
Demonstration by the scalar product [ modifier | Modifier and code ]
This property is a simple case of reducing the scalar function of Leibniz: just bring the point I in both vectors
And
, by the Chasles relationship:
We develop:
Point I is the middle of [ BC ], SO
And
are opposed, which implies that scalar products are eliminated and IC 2 = Single 2 SO
Demonstration using only theorems on distances [ modifier | Modifier and code ]
Either H the foot of the height from A . The three triangles Ahb , AHC And Raid are rectangles in H ; By applying the Pythagoras theorem to them, we get:
We can deduce :
We express HB And HC in terms of HI And WITH A . Even if it means intervening B And C If necessary, we can always assume that B And H are on the same side of I . SO,
We can therefore transform, in the expression above of
, under-expression
By replacing, we get:
Generalization to any Cévienne [ modifier | Modifier and code ]
The above demonstration by the scalar product is generalized, which makes it possible to demonstrate:
Be ( ABC ) a triangle, J A point of [ BC ] different from B , And k = JC / JB . SO :
Demonstration
Point J is the barium of ( B , k ) And ( C , 1) So ( cf. Scalar function of Leibniz)
Second median theorem – Be ( ABC ) a triangle and I The middle of the segment [ BC ]. SO
The demonstration uses the same decomposition of the vectors
And
that above:
Median theorem for a rectangle triangle [ modifier | Modifier and code ]
There is a special case relating to the rectangle triangle.
Theorem – In a rectangle triangle, the length of the median from the top of the right angle is half the length of the hypotenuse.
This theorem has a reciprocal.
Theorem – If in a triangle, the length of the median from a summit is half the length on the opposite side, then this triangle is rectangle on this summit.
Third median theorem – Be ( ABC ) a triangle and I The middle of the segment [ BC ]. On note H The orthogonal projected of A on ( BC ). SO
More precisely :
Or BC And THEM designate algebraic measures compared to the same unit director of the right ( BC ).
Just use the scalar product and remarkable identities:
The projection of
on ( BC ) East
from where
Stewart Theorem
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