Let P be an interior point of a triangle of area T. Through the point P, draw lines parallel to the three sides, partitioning the triangle into three triangles and three parallelograms. Let a, b and c be the areas of the three triangles. Prove that √T = √a + √b + √c.
The three triangles are similar to the triangle of area T (=each set of three angles, are the same for each little triangle and for the big one)
For easyness I'll consider here an isosceles triangle, but if it is not you can use the theorem of sines to related both different sides, the result will be the same)
We call L the repeated side of the big triangle and m, n, p, the correlative sides of each little triangles. We call z the angle from both sides L of the isosceles triangle.
The Area of the big triangle is then:
T=1/2*L^2 sin z
Then
sqr T= (sqr 2)/2 * L * sqr (sin z).
But L=m+n+p (you see this from geometry)
So sqr T= (sqr 2)/2 * m * sqr (sin z) + (sqr 2)/2 * n * sqr (sin z) + (sqr 2)/2 * p * sqr (sin z) = sqr a + sqr b + sqr c
Edited on January 29, 2019, 8:50 am
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Posted by armando
on 2019-01-29 03:54:11 |
Solution