Kansas University quarterly . 46 KANSAS UNIVERSITY QUARTERLY,. being transformed to P^. The cross-ratio of the pencil throughthe vertex C is _ sin ACP sin ACP,_Pb k,-^C(ABPPj— sin BCP sin BCP Pa -^^ ^. But perpendiculars from P and P, on the sides of the triangle are proportional to the areas of the triangles of which they are the , Pb .P,b,_APAC. APaACHence ky= altitudes. Pa P,a, ,PBC APiBC .\PAB APjAB ,, AtPBC APiBC In 1 ke manner kv=-~^ : ^=^^^— ; and k,—^ : -r-z—i . in like manner Kx ^p^c APjAC • —APAB APiAB We easily verify that kxkyl</,= i. But P and P, were taken to be any pa
Kansas University quarterly . 46 KANSAS UNIVERSITY QUARTERLY,. being transformed to P^. The cross-ratio of the pencil throughthe vertex C is _ sin ACP sin ACP,_Pb k,-^C(ABPPj— sin BCP sin BCP Pa -^^ ^. But perpendiculars from P and P, on the sides of the triangle are proportional to the areas of the triangles of which they are the , Pb .P,b,_APAC. APaACHence ky= altitudes. Pa P,a, ,PBC APiBC .\PAB APjAB ,, AtPBC APiBC In 1 ke manner kv=-~^ : ^=^^^— ; and k,—^ : -r-z—i . in like manner Kx ^p^c APjAC • —APAB APiAB We easily verify that kxkyl</,= i. But P and P, were taken to be any pair of corresponding points in the plane. Remembering the theorem* that the cross-ratio of the invariant elements and any pair of corresponding elements in a one-dimensional projective transformation is constant for all pairs of corresponding elements, we have found the following important theorem: 77/rorr/// 2. — llic cross-ratio of flw arras oj Jour triaiii^lcs ic/iosc vertices arc any pair of corrcspo/idi/ii:^ points in tlic transfor)nation T
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Keywords: ., bookcentury1800, bookdecade1890, bookpublisherlawrencekansastheu
