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Given a Pythagorean triple (a,b,c), the fractions a/b and b/a are called Pythagorean fractions. Diophantus showed that the Pythagorean fractions consist precisely of ...

A Pythagorean triangle is a right triangle with integer side lengths (i.e., whose side lengths (a,b,c) form a Pythagorean triple). A Pythagorean triangle with GCD(a,b,c)=1 is ...

A triangle center is regular iff there is a triangle center function which is a polynomial in Delta, a, b, and c (where Delta is the area of the triangle) such that the ...

Specifying two sides and the angle between them uniquely (up to geometric congruence) determines a triangle. Let c be the base length and h be the height. Then the area is ...

Specifying three sides uniquely determines a triangle whose area is given by Heron's formula, K=sqrt(s(s-a)(s-b)(s-c)), (1) where s=1/2(a+b+c) (2) is the semiperimeter of the ...

The Schoute center is the inverse of the symmedian point in the circumcircle. It has triangle center function alpha_(187)=a(2a^2-b^2-c^2) and corresponds to Kimberling center ...

In any triangle, the locus of a point whose pedal triangle has a constant Brocard angle and is described in a given direction is a circle of the Schoute coaxal system.

Let R+B be the number of monochromatic forced triangles (where R and B are the number of red and blue triangles) in an extremal graph. Then R+B=(n; 3)-|_1/2n|_1/4(n-1)^2_|_|, ...

The first mid-arc point is the triangle center with triangle center function alpha_(178)=[cos(1/2B)+cos(1/2C)]csc(1/2A). It is Kimberling center X_(178).

The second Morley cubic is the triangle cubic with trilinear equation It passes through Kimberling centers X_n for n=1, 1134, 1135, 1136, and 1137.