Arctic circle (geometry)

Polar circle

4 circle centers on a straight line:
arctic circle (d). Circumference (e), Feuerbachreis (t), circumference of the tangent triangle (s)

The Arctic Circle is a special circle of obtuse triangles . It is defined as the circle whose center corresponds to the vertical intersection and whose squared radius corresponds to the product of the length of the line from the vertical intersection to the height base and the vertical intersection to the corner.

For a triangle with an obtuse angle in , height intersection and height base points , and the corresponding polar circle has the center and the radius ${\ displaystyle \ triangle ABC}$ ${\ displaystyle \ alpha}$ ${\ displaystyle A}$ ${\ displaystyle H}$ ${\ displaystyle F_ {a}}$ ${\ displaystyle F_ {b}}$ ${\ displaystyle F_ {c}}$ ${\ displaystyle H}$

{\ displaystyle {\ begin {aligned} r_ {p} & = {\ sqrt {| HF_ {a} | \ cdot | HA |}} \\ & = {\ sqrt {| HF_ {b} | \ cdot HB | }} \\ & = {\ sqrt {| HF_ {c} | \ cdot | HC |}} \ end {aligned}}}

The height base points of the triangle are obtained by mirroring its corner points at the polar circle and vice versa. The radius of the arctic circle can also be calculated using the following two formulas:

{\ displaystyle {\ begin {aligned} r_ {p} & = {\ sqrt {-4R ^ {2} \ cos (\ alpha) \ cos (\ beta) \ cos (\ gamma)}} \\ & = { \ sqrt {4R ^ {2} - {\ frac {1} {2}} (a ^ {2} + b ^ {2} + c ^ {2})}} \ end {aligned}}}

For acute-angled triangles , the expression under the square root becomes negative and for right -angled triangles it becomes 0, so that the formula shows whether an obtuse triangle is present or whether a polar circle is defined.

The center of the polar circle lies with three other circles associated with the triangle on a common straight line, namely the circumference , the Feuerbach circle and the circumference of the tangent triangle of . In addition, the Feuerbach circle and the periphery emerge from a reflection at the polar circle. ${\ displaystyle \ triangle ABC}$ ${\ displaystyle \ triangle ABC}$

literature

HSM Coxeter, SL Greitzer: Simson Lines . §2.5 in Geometry Revisited. In: Math. Assoc. Amer., Washington DC 1967, pp. 136-138
Roger A. Johnson: Advanced Euclidean Geometry . Dover 2007, ISBN 978-0-486-46237-0 , pp. 176-181 (first published in 1929 by the Houghton Mifflin Company (Boston) under the title Modern Geometry )
Nathan Altshiller-Court: College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle . Dover, 2012 (Reprint), ISBN 9780486141374 , pp. 182-183

Web links

Eric W. Weisstein : Polar Circle . In: MathWorld (English).
Eckart Schmidt: All about the polar circle