Triangle of these conditions hold: (a) each of the angles is positive, and (b) the angles sum to 180°. If degenerate triangles are permitted, angles of 0° are permitted. Three positive angles "α", "β", and "γ", each of them less than 180°, are the angles of a triangle if and only if any one of the following conditions holds: the last equality applying only if none of the angles is 90° (so the tangent function's value is always finite). There are thousands of different constructions that find a special point associated with (and often inside) a triangle, satisfying some unique property:
Ailles rectangle between the two, with sides of lengths 2, 2 and (by the Pythagorean theorem) 2. The remaining space at the top of the rectangle is a right triangle with small angles of 15° and 75°. and sides of − 1, + 1 and 2. From the construction of the rectangle, it follows that and An alternative construction (also by Ailles) places a 30°–60°–90° triangle in the middle, flanked by two 45°–45°–90° triangles in the lower corners. The lower-left triangle has sides of 1, 1, and , while the lower-right triangle has sides of , , and . The 30°–60°–90° triangle