the square coincides with part of a side of the triangle). In a right triangle two of the squares coincide and have a vertex at the triangle's right angle, so a right triangle has only two distinct inscribed squares. An obtuse triangle has only one inscribed square, with a side coinciding with part of the triangle's longest side. Within a given triangle, a longer common side is associated with a smaller inscribed square. If an inscribed square has side of length qa and the triangle has a side of length a, part of which side coincides with a side of the square, then qa, a, the altitude ha from the side a, and the triangle's area T are related according to The largest possible ratio of the area of the inscribed square to the area of the triangle is 1/2, which occurs when , , and the altitude of the triangle from the base of length a is equal to a. The smallest possible ratio of the side of one inscribed square to the side of another in the same non-obtuse triangle is Both of these extreme cases occur for the isosceles right triangle. Triangles From an interior point in a reference triangle, the nearest points on the three sides serve as the vertices of the pedal triangle of that point. If the interior point is the circumcenter of the reference triangle, the vertices of the pedal triangle are the midpoints of the reference triangle's sides, and so the pedal triangle is called the midpoint triangle or medial triangle. The midpoint triangle subdivides the reference triangle into four congruent triangles which are similar to the reference triangle. The Gergonne triangle or intouch triangle of a reference triangle has its vertices at the three points of tangency of the reference triangle's sides with its incircle. The extouch triangle of a reference triangle has its vertices at the points of tangency of the reference triangle's excircles with its sides (not extended). Figures circumscribed about a triangle The tangential triangle of a reference triangle (other than a right triangle) is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at its vertices. As mentioned above, every triangle has a unique circumcircle, a circle passing through all three vertices, whose center is the intersection of the perpendicular bisectors of the triangle's sides. Further, every triangle has a unique Steiner