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PROPOSITION XLV. PROBLEM. To describe a parallelogram, which shall be equal to a given rectilinear figure, and have one of its angles equal to a given angle.
K Let ABCD be the given rectil. figure, and E the given L.
It is required to describe a O = to ABCD, having one of its Zs= . E.
Join AC. Describe a O FGHK = = 1 ABC, having - FKH= L E.
I. 42. To GH apply a O GHML= A CDA, having _GHM= 2 E.
I. 44. Then FKML is the reqd. For :: LGHM and 2 FKH are each= LE;
.:: ZGHM=. FKH, .. sum of 28 GHM, GHK=
=sum of 28 FKH, GHK =two rt. 28;
I. 29. .. KHM is a st. line.
I. 14. Again, :: HG meets the ||s FG, KM,
LFGH= LGHM, .. sum of us FGH, LGH=sum of 2s GHM, LGH
=two rt. 28;
I. 29. .. FGL is a st. line.
I. 14. Then :: KF is 1 to HG, and HG is I to LM .:. KF is || to LM;
I. 30. and KM has been shewn to be || to FL,
.: FKML is a parallelogram, and :: FH= A ABC, and GM: ACDA,
.:: O FM=whole rectil. fig. ABCD, and O FM has one of its ZS, FKM=.E. In the same way a O may be constructed equal to a given rectil. fig. of any number of sides, and having one of its angles equal to a given angle.
Q. Is F.
1. If one diagonal of a quadrilateral bisect the other, it divides the quadrilateral into two equal triangles.
2. If from any point in the diagonal, or the diagonal produced, of a parallelogram, straight lines be drawn to the opposite angles, they will cut off equal triangles.
3. In a trapezium the straight line, joining the middle points of the parallel sides, bisects the trapezium.
4. The diagonals AC, BD of a parallelogram intersect in 0, and P is a point within the triangle AOB ; prove that the difference of the triangles APB, CPD is equal to the sum of the triangles APC, BPD.
5. If either diagonal of a parallelogram be equal to a side of the figure, the other diagonal shall be greater than any side of the figure.
6. If through the angles of a parallelogram four straight lines be drawn parallel to its diagonals, another parallelogram will be formed, the area of which will be double that of the original parallelogram.
7. If two triangles have two sides respectively equal and the included angles supplemental, the triangles are equal.
8. Bisect a given triangle by a straight line drawn from a given point in one of the sides.
9. If the base of a triangle ABC be produced to a point D such that BD is equal to AB, and if straight lines be drawn from A and D to E, the middle point of BC; prove that the triangle ADE is equal to the triangle ABC.
10. Prove that a pair of the diagonals of the parallelograms, which are about the diameter of any parallelogram, are parallel to each other.
PROPOSITION XLVI, PROBLEM.
To describe a square upon a given straight line.
Let AB be the given st. line.
From A draw ACI to AB. I. 11. Cor.
In AC make AD=AB.
I. 31. Then AE is a parallelogram, and.. AB=ED, and AD=BE.
I. 34. But AB= AD;
.. AB, BE, ED, DA are all equal;
..AE is equilateral.
Def. xxx. and it is described on AB.
Q. E. P.
Ex. 1. Shew how to construct a rectangle whose sides are equal to two given straight lines.
Ex. 2. Shew that the squares on equal straight lines are equal.
Ex. 3. Shew that equal squares must be on equal straight lines.
Note. The theorems in Ex. 2 and 3 are assumed by Euclid in the proof of Prop. XLVIII.
PROPOSITION XLVII. THEOREM. In any right-angled triangle the square which is described on the side subtending the right angle is equal to the squares described on the sides which contain the right angle.
Let ABC be a right-angled 1, having the rt. BAC.
Then must sq. on BC= sum of sqq. on BA, AC. On BC, CA, AB descr. the sqq. BDEC, CKHA, AGFB. Through A draw AL || to BD or CE, and join AD, FC. Then :: _BAC and 2 BAG are both rt. 28, .. CAG is a st. line ;
I. 14. and :: _BAC and _CAH are both rt. 28; ..BAH is a st. line.
I. 14. Now ::: DBC= FBA, each being a rt. L,
adding to each ABC, we have
Ax. 2. Then in As ABD, FBC, :: AB=FB, and BD=BC, and LABD= . FBC, .:. Δ ABD=Δ FBC.
I. 4. Now O BL is double of A ABD, on same base BD and between same s AL, BD,
I. 41. and sq. BG is double of A FBC, on same base FB and bemeen same ||s FB, GC;
I. 41. ..O BL=sq. BG.
Similarly, by joining AE, BK it may be shewn that
O CL=sq. AK.
=sum of sq. BG and sq. AK,
Q. E. D.
Ex. 1. Prove that the square, described upon the diagonal of any given square, is equal to twice the given square.
Ex. 2. Find a line, the square on which shall be equal to the sum of the squares on three given straight lines.
Ex. 3. If one angle of a triangle be equal to the sum of the other two, and one of the sides containing this angle being divided into four equal parts, the other contains three of those parts ; the remaining side of the triangle contains five such parts.
Ex. 4. The triangles ABC, DEF, having the angles ACB, DHE right angles, have also the sides AB, AC equal to DE, DF, each to each ; shew that the triangles are equal in every respect.
NOTE. This Theorem has been already deduced as a Corollary from Prop. E, page 43.
Ex. 5. Divide a given straight line into two parts, so that the square on one part shall be double of the square on the other.
Ex. 6. If from one of the acute angles of a right-angled triangle a line be drawn to the opposite side, the squares on that side and on the line so drawn are together equal to the sum of the squares on the segment adjacent to the right angle and on the hypotenuse.
Ex. 7. In any triangle, if a line be drawn from the vertex at right angles to the base, the difference between the squares on the sides is equal to the difference between the squares on the segments of the base.