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Any two polar circles of two triangles in an orthocentric system are orthogonal. [1]: p. 177 The polar circles of the triangles of a complete quadrilateral form a coaxal system. [1]: p. 179 The most important property of the polar circle is the triangle is self-polar; the polar of each side/point is the opposite side/point.
If two lines a and k pass through a single point Q, then the polar q of Q joins the poles A and K of the lines a and k, respectively. The concepts of a pole and its polar line were advanced in projective geometry. For instance, the polar line can be viewed as the set of projective harmonic conjugates of a given point, the pole, with respect to ...
In the case that the side c has length π /2 on the unit sphere the equations governing the remaining sides and angles may be obtained by applying the rules for the right spherical triangle of the previous section to the polar triangle A'B'C' with sides a', b', c' such that A' = π − a, a' = π − A etc.
On Earth, the Arctic Circle is currently drifting northwards at a speed of about 14.5 m per year and is now at a mean latitude (i.e. without taking into account the astronomical nutation) of 66°33′50.3″ N; the Antarctic Circle is currently drifting southwards at a speed of about 14.5 m per year and is now at a mean latitude of 66°33′50. ...
The circle centered at H having radius the square root of this constant is the triangle's polar circle. [ 8 ] The sum of the ratios on the three altitudes of the distance of the orthocenter from the base to the length of the altitude is 1: [ 9 ] (This property and the next one are applications of a more general property of any interior point ...
The orthopole of lines passing through the circumcenter lie on the nine-point circle. A triangle's circumcircle, its nine-point circle, its polar circle, and the circumcircle of its tangential triangle [9] are coaxal. [10] Trilinear coordinates for the center of the Kiepert hyperbola are
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The area of a triangle is proportional to the excess of its angle sum over 180°. Two triangles with the same angle sum are equal in area. There is an upper bound for the area of triangles. The composition (product) of two reflections-across-a-great-circle may be considered as a rotation about either of the points of intersection of their axes.