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The following GEOMETRICAL PROBLEMS may be solved by simple applications of the Propositions in the First Two Books of Euclid. If the student prefer going on to the Third Book at once, he may do so ; but he will find his progress quicker, if he from time to time practises himself in these or any other Geometrical Problems.

Ir a perpendicular be drawn bisecting a given straight line, any point in this perpendicular is at equal distances, and any point without the perpendicular is at unequal distances from the extremities of the line.

Any side of a triangle is greater than the difference between the other two sides.

The difference of the angle at the base of any triangle is double the angle contained by a line drawn from the vertex perpendicular to the base, and another bisecting the angle at the vertex.

To find a point within a triangle which is equidistant from the three angles.

To describe a square which shall be equal to the difference of two squares, whose sides are given.

To describe a rectangular parallelogram which shall be equal to a given square, and have its adjacent sides together equal to a given line.

To describe a square which shall be equal to the sum of any number of given squares.

To inscribe a square in a given right-angled isosceles triangle.

To inscribe a square in a given quadrant of a circle.

If straight lines be drawn from the angles of a triangle through any point, either within or without the triangle, to meet the sides, and the lines joining these points of intersection and the sides of the triangle be produced to meet; the three points of concourse will be in the same straight line.

If the opposite sides or opposite angles of a quadrilateral figure be equal, the figure will be a parallelogram.

The two triangles, formed by drawing straight lines from any point within a parallelogram to the extremities of two opposite sides, are together half of the parallelogram.

If two sides of a trapezium be parallel, its area is equal to half that of a parallelogram whose base is the sum of those two sides, and altitude the perpendicular distance between them.

If from one of the acute angles of a right-angled triangle, a line be drawn to the opposite side; the squares of that side, and the line so drawn, are together equal to the squares of the segment adjacent to the right angle and of the hypo

tenuse.

In any triangle, if a line be drawn from the vertex at right angles to the base; the difference of the squares of the sides is equal to the difference of the squares of the segments of the base.

In any triangle, if a line be drawn from the vertex bisecting the base; the sum of the squares of the two sides of the triangle is double the sum of the squares of the bisecting line and of half the base.

120

THE ELEMENTS OF EUCLID.

Given one angle, a side adjacent to it, and the difference of the other two sides; to construct the triangle.

Given one angle, a side opposite to it, and the difference of the other two sides; to construct the triangle.

If the three sides of a triangle be bisected, the perpendiculars drawn to the sides at the three points of bisection, will meet in the same point

To trisect a given triangle from a given point within it.

To determine a point within a given triangle from which lines drawn to the several angles will divide the triangle into three equal parts.

Of all triangles having the same vertical angle, and whose bases pass through a given point, the least is that whose base is bisected in the given point.

The sum of the sides of an isosceles triangle is less than the sum of the sides of any other triangle on the same base and between the same parallels.

If from the extremity of the base of an isosceles triangle, a line equal to one of the sides be drawn to meet the opposite side; the angle formed by this line and the base produced, is equal to three times either of the equal angles of the triangle.

THE END.

LONDON:
SPOTTISWOODES and SHAW,
New-street-Square.

LONDON: 39, PATERNOSTER ROW:
MARCH 30, 1850.

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ELEMENTARY CREEK WORKS.

Yonge: An English-Greek Lexicon;

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