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PROPOSITION XXXIX. THEOREM.
Equal triangles upon the same base, and upon the same side of it, are between the same parallels.
Let the equal As ABC, DBC be on the same base BC, and on the same side of it.
Then must AD be || to BC.
For if not, through A draw AO || to BC, so as to meet BD, or BD produced, in O, and join OC.
Then : As ABC, OBC are on the same base and between the same ||s,
In the same way it
through A but AD is
.. AO is not || to BC.
may be shewn that no other line passing || to BC;
.. AD is || to BC.
Q. E. D.
Ex. 1. AD is parallel to BC; AC, BD meet in E; BC is produced to P so that the triangle PEB is equal to the triangle ABC: shew that PD is parallel to AC.
Ex. 2. If of the four triangles into which the diagonals divide a quadrilateral, two opposite ones are equal, the quadrilateral has two opposite sides parallel.
PROPOSITION XL. THEOREM.
Equal triangles upon equal bases, in the same straight line, and towards the same parts, are between the same parallels.
Let the equal As ABC, DEF be on equal bases BC, EF in the same st. line BF and towards the same parts.
Then must AD be || to BF.
For if not, through A draw 40 || to BF, so as to meet ED, or ED produced, in O, and join OF.
the less the greater, which is impossible.
.. AO is not || to BF.
In the same way it may be shewn that no other line passing through A but AD is || to BF,
.. AD is to BI.
Q. E. D.
Ex. 1. If the triangles be not towards the same parts, shew that the straight line joining the vertices of the triangles is bisected by the line containing the bases.
Ex. 2. The straight line, joining the points of bisection of two sides of a triangle, is parallel to the base.
Ex. 3. The straight lines, joining the middle points of the sides of a triangle, divide it into four equal triangles.
PROPOSITION XLI. THEOREM.
If a parallelogram and a triangle be upon the same base, and between the same parallels, the parallelogram is double of the triangle.
Let the ABCD and the A EBC be on the same base BC and between the same ||s AE, BC.
ABCD be double of ▲ EBC.
Then ▲ ABC= ▲ EBC,
between the same ||s;
and ABCD ;
they are on the same base and
ABCD is double of ▲ ABC, ·.· AC is a diagonal of
ABCD is double of ▲ EBC.
Q. E. D.
Ex. 1. If from a point, without a parallelogram, there be drawn two straight lines to the extremities of the two opposite sides, between which, when produced, the point does not lie, the difference of the triangles thus formed is equal to half the parallelogram.
Ex. 2. The two triangles, formed by drawing straight lines from any point within a parallelogram to the extremities of its opposite sides, are together half of the parallelogram.
To describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to a given angle.
Let ABC be the given ▲, and D the given
It is required to describe a equal to ▲ ABC, having one
of its 48=LD.
Bisect BC in E and join AE.
At E make CEF= L D.
Draw AFG || to BC, and from C draw CG | to EF.
Now ΔΑΕΒ= Δ AEC,
they are on equal bases and between the same [[s. .. AABC is double of ▲ AEC.
But FECG is double of ▲ AEC,
..they are on same base and between same [s.
:: □ FECG= ▲ ABC;
and FECG has one of its 4s, CEF= D.
Q. E F.
Ex. 1. Describe a triangle, which shall be equal to a given parallelogram, and have one of its angles equal to a given rectilineal angle.
Ex. 2. Construct a parallelogram, equal to a given triangle, and such that the sum of its sides shall be equal to the sum of the sides of the triangle.
Ex. 3. The perimeter of an isosceles triangle is greater than the perimeter of a rectangle, which is of the same altitude with, and equal to, the given triangle.
The complements of the parallelograms, which are about the diameter of any parallelogram, are equal to one another.
Let ABCD be a, of which BD is a diagonal, and EG, HK the Os about BD, that is, through which BD
and 'AF, FC the others, which make up the whole figure ABCD,
and which are.. called the Complements.
Then must complement AF-complement FC.
Hence sum of As HBF, EFD=sum of ▲s KFB, GDF.
Take these equals from ▲s ABD, CDB respectively, then remaining AF remaining FC.
Q. E. D.
Ex. 1. If through a point 0, within a parallelogram ABCD, two straight lines are drawn parallel to the sides, and the parallelograms OB, OD are equal; the point O is in the diagonal AC.
Ex. 2. ABCD is a parallelogram, AMN a straight line meeting the sides BC CD (one of them being produced) in M, N. Shew that the triangle MBN is equal to the triangle MDC.