Inscribed angle

Inscribed angle

Stress analysis, strain analysis, and shearing of soils

Published in Rodrigo Salgado, The Engineering of Foundations, Slopes and Retaining Structures, 2022

Mohr’s circles have interesting geometric properties. One very useful property that every circle has is that the central angle of the circle corresponding to a certain arc is twice as large as an inscribed angle corresponding to the same arc (Figure 4.5). Applying this property to the Mohr circle shown in Figure 4.3, the angle made by two straight lines drawn from any point of the circle to point S(σθ, τθ) representing the stresses on the plane of interest and to point S1(σ11, −σ13) is equal to θ. In particular, there is one and only one point P on the circle with the property that the line joining P to point S(σθ, τθ) is parallel to the plane on which σθ and τθ act. The point P with this property is known as the pole of the Mohr circle. Based on the preceding discussion, the pole can be defined as the point such that, if we draw a line through the pole parallel to the plane where stresses σθ and τθ act, this line intersects the Mohr circle at a point whose coordinates are σθ, τθ.

Focusing attention on auxiliary lines when introduced into geometric problems

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Journal Information

Published in International Journal of Mathematical Education in Science and Technology, 2019

Alik Palatnik, Avi Sigler

The auxiliary elements in the problem presented in Figure 9 are the centre, O, and the radii OA and OB, which extend to the endpoints of the inscribed angle. The auxiliary element concretizes the definition of a circle. The auxiliary element unites two previously unrelated angles by means of the central angle. and .

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Stress analysis, strain analysis, and shearing of soils

Focusing attention on auxiliary lines when introduced into geometric problems