Pythagoras-1

My OnPythagoras page demonstrates geometrically a2 + b2 = c2 for a right angled triangle whose sides have lengths a,b and c. Integer values of a,b and c form a Pythagorean triple. This page discusses Inverse triples.

Such a triangle for a given triple is shown with the line perpendicular to the hypotenuse at point D through the right angle corner at C. Its length d equals ab/c, as is shown by calculating the triangle's area and by similar triangles.

Lines BD and DA have lengths a2/c and b2/c.

Hence, by multiplying sides by c, all relevant lengths are integers, as shown on the right. This neatly shows a2 + b2 = c2.

One inverse formula is 1/a2 + 1/b2 = 1/d2.

If multiply all lengths by c, ie divide by c2, you get 1/(ac)2 + 1/(bc)2 = 1/(ab)2.

This has inverse squares of integers.

Use the 'explain' buttons to find out more.

Like WS Gilbert's Major General said, there are many useful facts about the square on the hypotenuse!