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If two perpendicular lines are drawn through the orthocenter H of any triangle, these lines intercept each side (or its extension) in two points (labeled P_(12), P_(12)^', ...

Let a convex cyclic polygon be triangulated in any manner, and draw the incircle to each triangle so constructed. Then the sum of the inradii is a constant independent of the ...

The 60 Pascal lines of a hexagon inscribed in a conic intersect three at a time through 20 Steiner points, and also three at a time in 60 points known as Kirkman points. Each ...

Consider a reference triangle DeltaABC and externally inscribe a square on the side BC. Now join the new vertices S_(AB) and S_(AC) of this square with the vertex A, marking ...

The answer to the question "which fits better, a round peg in a square hole, or a square peg in a round hole?" can be interpreted as asking which is larger, the ratio of the ...

The pentagonal hexecontahedron is the 60-faced dual polyhedron of the snub dodecahedron A_8 (Holden 1971, p. 55). It is Wenninger dual W_(18). A tetrahedron 10-compound, cube ...

The pentagonal icositetrahedron is the 24-faced dual polyhedron of the snub cube A_7 and Wenninger dual W_(17). The mineral cuprite (Cu_2O) forms in pentagonal ...

A number of attractive tetrahedron 10-compounds can be constructed. The first (left figures) can be obtained by combining tetrahedron 5-compounds of opposite chirality ...

In general, a tetrakis hexahedron is a non-regular icositetrahedron that can be constructed as a positive augmentation of a cube. Such a solid is also known as a ...

In general, a triakis tetrahedron is a non-regular dodecahedron that can be constructed as a positive augmentation of a regular tetrahedron. Such a solid is also known as a ...