and c, see here. The altitude from either leg coincides with the other leg. Since these intersect at the right-angled vertex, the right triangle's orthocenter—the intersection of its three altitudes—coincides with the right-angled vertex. Pythagorean theorem The Pythagorean theorem states that: In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle). This can be stated in equation form as where c is the length of the hypotenuse, and a and b are the lengths of the remaining two sides. Pythagorean triples are integer values of a, b, c satisfying this equation. Inradius and circumradius The radius of the incircle of a right triangle with legs a and b and hypotenuse c is The radius of the circumcircle is half the length of the hypotenuse, Thus the sum of the circumradius and the inradius is half the sum of the legs: One of the legs can be expressed in terms of the inradius and the other leg as Characterizations A triangle ABC with sides , semiperimeter , area T, altitude h opposite the longest side, circumradius R, inradius r, exradii ra, rb, rc (tangent to a, b, c respectively), and medians ma, mb, mc is a right triangle if and only if any one of the statements in the following six categories is true. Each of them is thus also a property of any right triangle. Sides and semiperimeter Angles A and B are complementary. Area where P is the tangency point of the incircle at the longest side AB. Inradius and exradii Altitude and medians The length of one median is equal to the circumradius. The shortest altitude (the one from the vertex with the biggest angle) is the geometric mean of the line segments it divides the opposite (longest) side into. This is the right triangle altitude theorem. Circumcircle and incircle The triangle can be inscribed in a semicircle, with one side coinciding with the entirety of the diameter (Thales' theorem). The circumcenter is the midpoint of the longest side. The longest side is a diameter of the circumcircle The circumcircle is tangent to the nine-point circle. The orthocenter lies on the circumcircle. The distance between the incenter and the orthocenter is equal