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Let five circles with concyclic centers be drawn such that each intersects its neighbors in two points, with one of these intersections lying itself on the circle of centers. ...

Let two disks of radius r intersect one another perpendicularly and have a diameter in common. If the distance between the centers of the disks is sqrt(2) times their radius, ...

Let S be a subset of a metric space. Then the set S is open if every point in S has a neighborhood lying in the set. An open set of radius r and center x_0 is the set of all ...

Given an obtuse triangle, the polar circle has center at the orthocenter H. Call H_i the feet. Then the square of the radius r is given by r^2 = HA^_·HH_A^_ (1) = HB^_·HH_B^_ ...

Let C be a curve and let O be a fixed point. Let P be on C and let Q be the curvature center at P. Let P_1 be the point with P_1O a line segment parallel and of equal length ...

A configuration of 12 planes and 12 points such that six points lie in every plane and six planes pass through every point. Alternatively, the configuration consists of 16 ...

One of the three standard tori given by the parametric equations x = (c+acosv)cosu (1) y = (c+acosv)sinu (2) z = asinv (3) with c>a. This is the torus which is generally ...

A rounded rectangle is the shape obtained by taking the convex hull of four equal circles of radius r and placing their centers at the four corners of a rectangle with side ...

There are at least two theorems known as Salmon's theorem. This first states that if P and S are two points, PX and SY are the perpendiculars from P and S to the polars of S ...

The second Brocard circle is the circle having center at the circumcenter O of the reference triangle and radius R_B = sqrt(1-4sin^2omega)R (1) = (2) where R is the ...