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" Every straight line cutting the sides of a triangle (produced when necessary) determines upon the sides six segments, such that the product of three non-consecutive segments is equal to the product of the other three. "
Exercise Manuals - Page 65
by George Albert Wentworth - 1889
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Wentworth & Hill's Exercise Manuals: Geometry, Issue 3

George Albert Wentworth - 1884 - 264 pages
...1678. cc Fig. 60. 40. Conversely, if three lines drawn through the vertices of a triangle determine upon the sides six segments, such that the product...non-consecutive segments is equal to the product of the three others, the lines pass through the same point. The proof is similar to that of Ex. 38. 41. By...
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Wentworth & Hill's Exercise Manuals, Issue 3

George Albert Wentworth - 1886 - 264 pages
...PAM&nA PCR are similar ; therefore, PA : PM = PC ': PS. But PA:PM= PN: PB. .-.PN:PB - PC:PR, and PNx PŁ 36. The common exterior tangents to two circles pass...same. Draw CD II to AB. From the similar triangles, = CD CZ Therefore, AXxBY . CY CD AZxBX. CYx CD CZx CD whence, AXxBYxCZ=AZx BXX CY. This theorem was...
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Wentworth & Hills's Exercise Manuals: Geometry, Issue 3

George Albert Wentworth - 1889 - 264 pages
...AP=AZ; 39. Lines drawn through the vertices of a triangle, and passing through a common point, determine upon the sides six segments, such that the product...non-consecutive segments is equal to the product of the three others. The common point 0 may lie either within the triangle (Fig. 59), or without the triangle...
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Original Exercises in Plane and Solid Geometry

Levi Leonard Conant - Geometry - 1905 - 134 pages
...diagonals and nse No. 521. 525. Every straight line cutting the sides of a triangle, or the sides produced, determines upon the sides six segments such that the...segments is equal to the product of the other three. HINT. The cutting line DEF may cut two sides of the triangle and the third side produced, or the three...
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School Science and Mathematics, Volume 9

Education - 1909 - 1030 pages
...DS:YP:CQ=CS:DP:YQ ................................. (2) Since a line cutting the sides of a triangle determines upon the sides six segments, such that...segments is equal to the product of the other three. Also from the similar triangles APY and DPX, we get, AP :DP= XP :YP ......................................................
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School Science and Mathematics, Volume 7

Education - 1907 - 880 pages
...and in triangle CED, YD -LC BZ = BY. LD. ZC ...... (2), since a line cutting the sides of a triangle determines upon the sides six segments, such that...segments is equal to the product of the other three. Also, the triangles BEZ and AEY are similar, respectively to triangles CZF and DYF. Therefore, BZ :...
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School Science and Mathematics, Volume 7

Education - 1879 - 944 pages
...in triangle CBD, YD . LC EZ = EY . LD . ZC ...... (2), since a line cutting the sides of a triangle determines upon the sides six segments, such that...segments is equal to the product of the other three. Also, the triangles.BEZ and AEY are similar, respectively to triangles CZF and DYF. Therefore, BZ :...
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School Science and Mathematics, Volume 21

Education - 1921 - 970 pages
...1678) : Lines drawn through the vertices of a triangle, and passing through a common point, determine upon the sides six segments, such that the product...non-consecutive segments is equal to the product of the three other segments. I. Solution by Michael Goldberg, Philadelphia, I'n. Let ABC be the triangle,...
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Original Exercises in Plane and Solid Geometry

Levi Leonard Conant - Geometry - 1905 - 140 pages
...through the vertices of a triangle determine, if concurrent, six segments on the sides of the triangle such that the product of three non-consecutive segments is equal to the product of the other three. HINT. The lines may meet within or without the triangle. Use No. 525 on each of the two parts into...
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School Science and Mathematics, Volume 20

Education - 1920 - 928 pages
...triangle, and passing through a common point, determine upon the sides six segments, such that tbe product of three non-consecutive segments is equal to the product of the three other segments. 678. Proposed by A. Pelletier, Ecole Polylechnique, Montreal, Can. Prove that...
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