PROPOSITION 3.

If a straight line be cut in extreme and mean ratio, the square on the lesser segment added to the half of the greater segment is five times the square on the half of the greater segment.

For let any straight line AB be cut in extreme and mean ratio at the point C, let AC be the greater segment, and let AC be bisected at D ; I say that the square on BD is five times the square on DC.

For let the square AE be described on AB, and let the figure be drawn double.

A D C B R G O P Q M H K F N L S E
Since AC is double of DC,
therefore the square on AC is quadruple of the square on DC, that is, RS is quadruple of FG.
And, since the rectangle AB, BC is equal to the square on AC, and CE is the rectangle AB, BC,
therefore CE is equal to RS.
But RS is quadruple of FG ;
therefore CE is also quadruple of FG.
Again, since AD is equal to DC,
HK is also equal to KF.
Hence the square GF is also equal to the square HL.
Therefore GK is equal to KL, that is, MN to NE ; hence MF is also equal to FE.
But MF is equal to CG ;
therefore CG is also equal to FE.
Let CN be added to each ;
therefore the gnomon OPQ is equal to CE.
But CE was proved quadruple of GF ;
therefore the gnomon OPQ is also quadruple of the square FG.
Therefore the gnomon OPQ and the square FG are five times FG.
But the gnomon OPQ and the square FG are the square DN.
And DN is the square on DB, and GF the square on DC.
Therefore the square on DB is five times the square on DC.
Q. E. D.

Dependency Graph

digraph { bgcolor="transparent"; node [shape=box, style="rounded,filled", fontname="Helvetica", fontsize=11, color="#d6d3cd", fontcolor="#ffffff"]; edge [color="#8c857b", fontcolor="#8c857b", arrowsize=0.75]; rankdir="TB"; "I.def.1" [label="I.def.1", fillcolor="#224422", tooltip="I.def.1: A point is that which has no part.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-1", target="_top"]; "I.def.2" [label="I.def.2", fillcolor="#224422", tooltip="I.def.2: A line is breadthless length.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-2", target="_top"]; "I.def.3" [label="I.def.3", fillcolor="#224422", tooltip="I.def.3: The extremities of a line are points.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-3", target="_top"]; "I.def.4" [label="I.def.4", fillcolor="#224422", tooltip="I.def.4: A straight line is a line which lies evenly with the points on itself.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-4", target="_top"]; "I.def.5" [label="I.def.5", fillcolor="#224422", tooltip="I.def.5: A surface is that which has length and breadth only.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-5", target="_top"]; "I.def.6" [label="I.def.6", fillcolor="#224422", tooltip="I.def.6: The extremities of a surface are lines.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-6", target="_top"]; "I.def.7" [label="I.def.7", fillcolor="#224422", tooltip="I.def.7: A plane surface is a surface which lies evenly with the straight lines on itself.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-7", target="_top"]; "I.def.8" [label="I.def.8", fillcolor="#224422", tooltip="I.def.8: A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-8", target="_top"]; "I.def.10" [label="I.def.10", fillcolor="#224422", tooltip="I.def.10: When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-10", target="_top"]; "I.def.13" [label="I.def.13", fillcolor="#224422", tooltip="I.def.13: A boundary is that which is an extremity of anything.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-13", target="_top"]; "I.def.14" [label="I.def.14", fillcolor="#224422", tooltip="I.def.14: A figure is that which is contained by any boundary or boundaries.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-14", target="_top"]; "I.def.15" [label="I.def.15", fillcolor="#224422", tooltip="I.def.15: A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-15", target="_top"]; "I.def.16" [label="I.def.16", fillcolor="#224422", tooltip="I.def.16: And the point is called the centre of the circle.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-16", target="_top"]; "I.def.17" [label="I.def.17", fillcolor="#224422", tooltip="I.def.17: A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-17", target="_top"]; "I.def.19" [label="I.def.19", fillcolor="#224422", tooltip="I.def.19: Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-19", target="_top"]; "I.def.20" [label="I.def.20", fillcolor="#224422", tooltip="I.def.20: Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-20", target="_top"]; "I.def.22" [label="I.def.22", fillcolor="#224422", tooltip="I.def.22: Of quadrilateral figures, a square is that which is both equilateral and right-angled ; an oblong that which is right-angled but not equilateral ; a rhombus that which is…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-22", target="_top"]; "II.def.1" [label="II.def.1", fillcolor="#224422", tooltip="II.def.1: Any rectangular parallelogram is said to be contained by the two straight lines containing the right angle.", URL="../../../../volume_I/book_ii/definitions/index.html#ii-def-1", target="_top"]; "II.def.2" [label="II.def.2", fillcolor="#224422", tooltip="II.def.2: And in any parallelogrammic area let any one whatever of the parallelograms about its diameter with the two complements be called a gnomon.", URL="../../../../volume_I/book_ii/definitions/index.html#ii-def-2", target="_top"]; "VI.def.3" [label="VI.def.3", fillcolor="#224422", tooltip="VI.def.3: A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.", URL="../../../../volume_II/book_vi/definitions/index.html#vi-def-3", target="_top"]; "XIII.3" [label="XIII.3", penwidth=2.5, color="#e8b85a", fillcolor="#4a453b", tooltip="XIII.3: If a straight line be cut in extreme and mean ratio, the square on the lesser segment added to the half of the greater segment is five times the square on the half of the greater…", URL="index.html", target="_top"]; "I.def.3" -> "I.def.1"; "I.def.4" -> "I.def.1"; "I.def.15" -> "I.def.1"; "I.def.16" -> "I.def.1"; "I.def.3" -> "I.def.2"; "I.def.4" -> "I.def.2"; "I.def.6" -> "I.def.2"; "I.def.8" -> "I.def.2"; "VI.def.3" -> "I.def.2"; "I.def.4" -> "I.def.3"; "I.def.13" -> "I.def.3"; "I.def.7" -> "I.def.4"; "I.def.8" -> "I.def.4"; "I.def.10" -> "I.def.4"; "I.def.15" -> "I.def.4"; "I.def.17" -> "I.def.4"; "I.def.19" -> "I.def.4"; "II.def.1" -> "I.def.4"; "XIII.3" -> "I.def.4"; "I.def.6" -> "I.def.5"; "I.def.7" -> "I.def.5"; "I.def.15" -> "I.def.6"; "I.def.8" -> "I.def.7"; "I.def.15" -> "I.def.7"; "I.def.10" -> "I.def.8"; "I.def.22" -> "I.def.8"; "II.def.1" -> "I.def.10"; "I.def.14" -> "I.def.13"; "I.def.15" -> "I.def.14"; "XIII.3" -> "I.def.14"; "I.def.16" -> "I.def.15"; "I.def.17" -> "I.def.15"; "I.def.17" -> "I.def.16"; "II.def.2" -> "I.def.17"; "I.def.20" -> "I.def.19"; "I.def.22" -> "I.def.19"; "I.def.22" -> "I.def.20"; "XIII.3" -> "I.def.22"; "XIII.3" -> "II.def.1"; "XIII.3" -> "II.def.2"; "XIII.3" -> "VI.def.3"; }

Required for

XIII.16, XIII.18

See Commentary on Proposition XIII.3.