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By William J. M'Clelland
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Additional info for A treatise on the geometry of the circle and some extensions to conic sections by the method of reciprocation, with numerous examples.
If the sides of the given escribed triangle be and a, /3, y the distances of from P, Q, R, yu, and i/, Xa + Hence required / u/3 + i>y = a minimum. find a point, given multiples of whose distances from three fixed points is a minimum when any to two of the multiples are together greater than the third. EXAMPLES. H d denote the distance of the point from the circum centre of the triangle ABC', prove that twice the area of the minimum 1. If triangle PQR is [For join join BC'. AO and (R 2 ~ d 2) produce sin it to A sin B sin C.
Absolute values only the equation 23. 14 + 12. 34 = This is is 31". 2 of great importance, and should be borne in in the following Examples. mind INTRODUCTION. 13 EXAMPLES. 1. What does the general property reduce to become points ? when the circles Ptolemy's Theorem. Express a condition that the circum-circle of a given triangle touch another circle. [If a, 6, c be the sides and t l9 t^ t3 the tangents from the vertices 2. may to the other circle 3. we have a^ 1 + Feuerbach's Theorem. ] The nine points circle of a triangle touches the in- and ex-circles.
If the sides of 6. Hart's Extension of Feuerbach's Theorem. a triangle be replaced by three circles, and four circles corresponding to the in- and ex-circles of the triangle described to touch them the group of four is touched by a circle. ; [Let the triangle formed by the circles be ABC, and let as b>s c. If the in- and ex-circles are numbered Then * This proof is an application of the converse of Dr. Casey's relation. t This result may be otherwise shown as follows Then EG of contact of the tangent.