Geometric mean theorem
The set group of Pythagoras includes three sets of mathematics dealing with calculations in right triangles:
- 2.1 Algebraic evidence 2.1.1 Proof of the amount set
- 2.1.2 Proof of Kathetensatzes
- 2.1.3 Proof of Kathetensatzes with the height set
- 2.1.4 Proof of the Pythagorean Theorem using the Kathetensatzes
- 2.2.1 Supplementary proof of the amount set
- 2.2.2 Scherungsbeweis
- 2.2.3 Scherungsbeweis of Kathetensatzes
- 2.2.4 Proof of the complete set of records about similar triangles
The individual records
Pythagorean Theorem
The Pythagorean theorem states that in a right-angled triangle the area of the large square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides.
As a formula:
Kathetensatz of Euclid
The point of the height h of the hypotenuse is divided into two parts, p and q. The ratio of these two parts is described by the Kathetensatz. He says that the rectangles that are each acreage equal to the hypotenuse under the Kathetenquadraten in right triangles in the square. or:
As formulas:
Height Type of Euclid
The height theorem says that in a right-angled triangle the square area equal to the square of the hypotenuse to the height. or:
The converse is also true:
Evidence
For the Pythagorean theorem exist very many different proofs, see Article theorem of Pythagoras. From this one can prove the amount set and the Kathetensatz by algebraic calculation, but also vice versa follows from each of these two sets of Pythagorean theorem! The three movements are therefore equivalent: If one of the three sets shown, as are the other two sets, the set group apply.
Algebraic proofs
Proof of the amount set
The proof of the level set can be performed with the Pythagorean theorem and the binomial formula.
In the diagram it can be seen the three right-angled triangles, one of the sides a, b , c, then each one with H, P, a and b, q, b. For each of the triangles of the Pythagorean theorem applies:
Moreover applies. The square is:
After the first binomial formula, this is
Substituting this for the first c ² in formula and A ² and B ² each left side of the second and third formula, we obtain:
And thus. After division by two of the amount to be proved theorem follows:
Proof of Kathetensatzes
This proof is analogous to the proof of the amount set using the above four formulas: It is
And thus
Similarly applies then
Proof of Kathetensatzes with the height set
Based on the graph in the proof of the amount set:
Proof of the theorem of Pythagoras with the Kathetensatzes
From the validity of Kathetensatzes but the Pythagorean theorem can be easily derived:
Geometric evidence
For the level set and the Kathetensatz geometrical proofs exist:
Supplementary evidence of the amount set
Supplementary evidence to the amount set
Two right triangles are congruent if the other sides are the same ( the included angle is indeed the same).
If you divide a right-angled triangle of height h into two right triangles with sides p and h and q and h ( yellow and red triangle in the diagram), so can these h on a square with sides of length ( in the diagram below left) and a rectangle with the sides p and q create ( in the diagram below right).
In both cases, a right-angled triangle with the short sides is formed p h q h The right and left triangle are therefore congruent. The first comprises but from the yellow and red triangle and the square h ², the second of the two triangles and the rectangle pq. The area of the square must therefore be equal to the area of the rectangle, ie, h ² = pq.
Scherungsbeweis
Shearing a rectangle to a parallelogram, the surface is maintained. This allows the level set also prove. The animation illustrates the proof:
Illustrate the proof passage for the amount set by shear
Using the congruence of triangles you have to prove yet that the new height q actually corresponds to the Hypotenusenabschnitt. It is omitted here.
Scherungsbeweis of Kathetensatzes
The Scherungsbeweis Pythagoras' theorem proves the same time the Kathetensatz.
Illustrate the proof passage for Kathetensatz sheared
Evidence of the complete set of records about similar triangles
The aspect ratios of similar triangles provide immediately the two Kathetensätze and the height set. The Pythagorean theorem then follows directly from the application of the two Kathetensätze.