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Given a triangle DeltaABC and the excentral triangle DeltaJ_AJ_BJ_C, define the A^'-vertex of the hexyl triangle as the point in which the perpendicular to AB through the ...

Given two circles, draw the tangents from the center of each circle to the sides of the other. Then the line segments AB and CD are of equal length. The theorem can be proved ...

Integrals over the unit square arising in geometric probability are int_0^1int_0^1sqrt(x^2+y^2)dxdy=1/3[sqrt(2)+sinh^(-1)(1)] int_0^1int_0^1sqrt((x-1/2)^2+(y-1/2)^2)dxdy ...

An isosceles tetrahedron is a nonregular tetrahedron in which each pair of opposite polyhedron edges are equal, i.e., a^'=a, b^'=b, and c^'=c, so that all triangular faces ...

Marion's theorem (Mathematics Teacher 1993, Maushard 1994, Morgan 1994) states that the area of the central hexagonal region determined by trisection of each side of a ...

The point of concurrence of the four maltitudes of a cyclic quadrilateral. Let M_(AC) and M_(BD) be the midpoints of the diagonals of a cyclic quadrilateral ABCD, and let P ...

A Cevian point P is point with respect to which a set of Cevians of a triangle are taken. Connecting the endpoints of the Cevians through a point P gives the so-called Cevian ...

Given positive numbers s_a, s_b, and s_c, the Elkies point is the unique point Y in the interior of a triangle DeltaABC such that the respective inradii r_a, r_b, r_c of the ...

In a given acute triangle DeltaABC, locate a point whose distances from A, B, and C have the smallest possible sum. The solution is the point from which each side subtends an ...

Three concurrent homologous lines pass respectively through three fixed points on the similitude circle which are known as the invariable points.