A treatise on the geometry of the circle and some extensions - download pdf or read online

By William J. M'Clelland

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Extra resources for A treatise on the geometry of the circle and some extensions to conic sections by the method of reciprocation, with numerous examples.

Example text

MAXIMUM AND MINIMUM. 22 The (Euc. since triangles I. 4) ; BGL hence I. (Euc. and B'CL are equal 20) in every respect = Similarly BP B'P. BC= B'C. AP + B'P > AB' Hence follows it that AP + BP> AC + BC. COR. 1. If the line difference of the sides L cuts the base internally the (ACBC) is a maximum when it bisects internally the angle G. EXAMPLES. The 1. triangle of minimum perimeter inscribed in a given one is Z of the perpendiculars * let fall formed by joining the feet A", T, from the vertices on the opposite sides.

22. When the line OP is perpendicular to EC, OQ and OR are therefore perpendiculars to CA and AB respecas centre and OP as radius tively, and the circle with touches BC. In this case the two solutions coincide, and PQR is the minimum triangle of given species that can other be inscribed 23. It is escribed to in ABC. manifest that a given triangle ABC may be For having determined the another PQR. point 0, the triangles species, and BOG, CO A, and AOB are given in are therefore completely determined, since For other solutions see " Mathematics from the Educational Times" Vol.

MAXIMUM AND MINIMUM. P on C Given an angle Problem. 11. point 23 of a triangle ike base, construct the triangle of and a minimum area. P draw APE such that AP = BP. triangle ABC is less than any other A'B'C. The Through AX For draw Then the triangles parallel to BB and BPB' are equal in all respects (Euc. I. 4); hence AA'P > BB'P. To each add APB'G, therefore f . APX A'B'C>ABC\ whose base hence the triangle of least area is that is bisected at this point. Given an angle and any curve concave The tangent AB which forms ivith the sides of the angle a triangle ABC of minimum area is 12.

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A treatise on the geometry of the circle and some extensions to conic sections by the method of reciprocation, with numerous examples. by William J. M'Clelland


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