Pythagorean theorem
Inverse Pythagorean theorem

Given a right triangle with sides a, b, c {\displaystyle a,b,c} and altitude d {\displaystyle d} (a line from the right angle and perpendicular to the hypotenuse c {\displaystyle c}). The Pythagorean theorem has,

while the inverse Pythagorean theorem relates the two legs a, b {\displaystyle a,b} to the altitude d {\displaystyle d},

The equation can be transformed to,

where x 2 + y 2 = z 2 {\displaystyle x^{2}+y^{2}=z^{2}} for any non-zero real x, y, z {\displaystyle x,y,z}. If the a, b, d {\displaystyle a,b,d} are to be integers, the smallest solution a > b > d {\displaystyle a>b>d} is then

using the smallest Pythagorean triple 3, 4, 5 {\displaystyle 3,4,5}. The reciprocal Pythagorean theorem is a special case of the optic equation

where the denominators are squares and also for a heptagonal triangle whose sides p, q, r {\displaystyle p,q,r} are square numbers.
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