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ContentElementary Geometry
Elementary GeometryMiscellaneous Propositions926 In a given line π΄πΆ, to find a point π whose distance from a point π shall have a given ratio to its distance in a given direction from a line π΄π΅.Through π draw π΅ππΆ parallel to the given direction. Produce π΄π, and make πΆπΈ in the given ratio to πΆπ΅. Draw ππ parallel to πΈπΆ, and ππ to πΆπ΅. There are two solutions when πΆπΈ cuts π΄π in two points. Proof. By VI. 2 927 To find a point π in π΄πΆ, whose distance ππ from π΄π΅ parallel to π΅πΆ shall have a given ratio to its distance ππ from π΅πΆ parallel to π΄π·. Draw π΄πΈ parallel to π΅πΆ, and having to π΄π· the given ratio. Join π΅πΈ cutting π΄πΆ in π, the point required. Proof. By VI. 2 928 To find a point π on any line, straight or curved, whose distances ππ, ππ, in given directions from two given lines π΄π, π΄π΅, shall be in a given ratio. Take π any point in the first line. Draw ππ΅ parallel to the direction of ππ, and π΅πΆ parallel to that of ππ, making ππ΅ have to π΅πΆ the given ratio. Join ππΆ, cutting π΄π΅ in π·. Draw π·πΈ parallel to πΆπ΅. Then π΄πΈ produced cuts the lines in π, the point required, and is the locus of such points. Proof. By VI. 2 929 To draw a line ππ through a given point π so that the segments ππ, ππ, intercepted by a given circle, shall be in a given ratio. Divide the radius of the circle in that ratio, and, with the parts for sides, construct a triangle ππ·πΆ upon ππΆ as base. Produce πΆπ· to cut the circle in π. Draw πππ and πΆπ. Then ππ·+π·πΆ=radius therefore ππ·=π·π But πΆπ=πΆπ therefore ππ· is parallel to πΆπ (I 5, 28) therefore β― by (VI. 2) 930 From a given point π in the side of a triangle, to draw a line ππ which shall divide the area of the triangle in a given ratio. Divide π΅πΆ in π· in the given ratio, and draw π΄π parallel to ππ·. ππ will be the line required. π΄π΅π·: π΄π·πΆ= the given ratio (VI. 1), and π΄ππ·=πππ· (I.37); therefore β― Sources and Referenceshttps://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdriveΒ©sideway ID: 210900016 Last Updated: 9/16/2021 Revision: 0 Ref: References
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