mr-HESHAM
03-12-2010, 11:21 PM
http://3.bp.blogspot.com/_Oc5DDdaxrtk/TO_3ODX_8BI/AAAAAAAAABw/Hx3kEvGhFIc/s1600/220px-Pythagoras_similar_triangles.PNG
Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. We draw the altitude from point C, and call H its intersection with the side AB. Point H divides the length of the hypotenuse c into parts d and e. The new triangle ACH is similar to triangle ABC, because they both have a right angle (by definition of the altitude), and they share the angle at A, meaning that the third angle will be the same in both triangles as well, marked as θ in the figure. By a similar reasoning, the triangle CBH is also similar to ABC. The proof of similarity of the triangles requires the Triangle postulate: the sum of the angles in a triangle is two right angles, and is equal to the parallel postulate. Similarity of the triangles leads to the equality of ratios of corresponding sides
:
http://upload.wikimedia.org/math/a/8/7/a87b70928af5ddc77b39959e3f00cf3f.png
The first result equates the cosine of each angle θ and the second result equates the sines.
These ratios can be written as:
http://upload.wikimedia.org/math/1/9/5/1954c4f419fd8de0b55d66e10bf0504f.png
Summing these two equalities, we obtain
http://upload.wikimedia.org/math/c/8/1/c81f0514c0c1ef55c5d9f52ccbb24cce.png
which, tidying up, is the Pythagorean theorem
:
http://upload.wikimedia.org/math/7/2/2/722ef5bc9fc654ba7f505f3db850030e.png
Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. We draw the altitude from point C, and call H its intersection with the side AB. Point H divides the length of the hypotenuse c into parts d and e. The new triangle ACH is similar to triangle ABC, because they both have a right angle (by definition of the altitude), and they share the angle at A, meaning that the third angle will be the same in both triangles as well, marked as θ in the figure. By a similar reasoning, the triangle CBH is also similar to ABC. The proof of similarity of the triangles requires the Triangle postulate: the sum of the angles in a triangle is two right angles, and is equal to the parallel postulate. Similarity of the triangles leads to the equality of ratios of corresponding sides
:
http://upload.wikimedia.org/math/a/8/7/a87b70928af5ddc77b39959e3f00cf3f.png
The first result equates the cosine of each angle θ and the second result equates the sines.
These ratios can be written as:
http://upload.wikimedia.org/math/1/9/5/1954c4f419fd8de0b55d66e10bf0504f.png
Summing these two equalities, we obtain
http://upload.wikimedia.org/math/c/8/1/c81f0514c0c1ef55c5d9f52ccbb24cce.png
which, tidying up, is the Pythagorean theorem
:
http://upload.wikimedia.org/math/7/2/2/722ef5bc9fc654ba7f505f3db850030e.png