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Elementary Geometry
 Miscellaneous Propositions
 Sources and References

Elementary Geometry

Miscellaneous Propositions

926 image In a given line 𝐴𝐢, to find a point 𝑋 whose distance from a point 𝑃 shall have a given ratio to its distance in a given direction from a line 𝐴𝐡.
Through 𝑃 draw 𝐡𝑃𝐢 parallel to the given direction. Produce 𝐴𝑃, and make 𝐢𝐸 in the given ratio to 𝐢𝐡. Draw 𝑃𝑋 parallel to 𝐸𝐢, and π‘‹π‘Œ to 𝐢𝐡. There are two solutions when 𝐢𝐸 cuts 𝐴𝑃 in two points.  Proof. By VI. 2 927 image To find a point 𝑋 in 𝐴𝐢, whose distance π‘‹π‘Œ from 𝐴𝐡 parallel to 𝐡𝐢 shall have a given ratio to its distance 𝑋𝑍 from 𝐡𝐢 parallel to 𝐴𝐷.
Draw 𝐴𝐸 parallel to 𝐡𝐢, and having to 𝐴𝐷 the given ratio. Join 𝐡𝐸 cutting 𝐴𝐢 in 𝑋, the point required. Proof. By VI. 2 928 image To find a point 𝑋 on any line, straight or curved, whose distances π‘‹π‘Œ, 𝑋𝑍, in given directions from two given lines 𝐴𝑃, 𝐴𝐡, shall be in a given ratio.
Take 𝑃 any point in the first line. Draw 𝑃𝐡 parallel to the direction of π‘‹π‘Œ, and 𝐡𝐢 parallel to that of 𝑋𝑍, making 𝑃𝐡 have to 𝐡𝐢 the given ratio. Join 𝑃𝐢, cutting 𝐴𝐡 in 𝐷. Draw 𝐷𝐸 parallel to 𝐢𝐡. Then 𝐴𝐸 produced cuts the lines in 𝑋, the point required, and is the locus of such points. Proof. By VI. 2 929 image To draw a line π‘‹π‘Œ through a given point 𝑃 so that the segments 𝑋𝑃, π‘ƒπ‘Œ, 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 𝑃𝐷𝐢 upon 𝑃𝐢 as base. Produce 𝐢𝐷 to cut the circle in 𝑋. Draw π‘‹π‘ƒπ‘Œ and πΆπ‘Œ.
Then 𝑃𝐷+𝐷𝐢=radius therefore 𝑃𝐷=𝐷𝑋 But πΆπ‘Œ=𝐢𝑋 therefore 𝑃𝐷 is parallel to πΆπ‘Œ (I 5, 28) therefore β‹― by (VI. 2) 930 image From a given point 𝑃 in the side of a triangle, to draw a line 𝑃𝑋 which shall divide the area of the triangle in a given ratio.
Divide 𝐡𝐢 in 𝐷 in the given ratio, and draw 𝐴𝑋 parallel to 𝑃𝐷. 𝑃𝑋 will be the line required.
𝐴𝐡𝐷: 𝐴𝐷𝐢= the given ratio (VI. 1), and 𝐴𝑃𝐷=𝑋𝑃𝐷 (I.37); therefore β‹―

Sources and References


ID: 210900016 Last Updated: 9/16/2021 Revision: 0 Ref:



  1. Hilbert, D. (translated by Townsend E.J.), 1902, The Foundations of Geometry
  2. Moore, E.H., 1902, On the projective axioms of geometry
  3. Fitzpatrick R. (translated), Heiberg J.L. (Greek Text), Euclid (Author), 2008, Euclid's Elements of Geometry

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