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Upon a given line A B to describe a square.

1. Upon A and B, as centres with a radius A B, describe two arcs A e C, Be D, cutting each other at e.

2. Bisect A e at f, from e make e D and e C equal to e f 3. Join A D, D C, C B, it will be the square required.

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Upon a given line A B, to construct any regular polygon.

1. Upon A and B, as centres with a radius A B, describe two arcs intersecting each other at F.

2. From B, draw B C perpendicular, and divide the arc AC into as many equal parts as the polygon is to have sides.

3. Through the second division D, draw B G, make F E equal to F D, and through E, draw A G, meeting B G at G, then G will be the centre, and G A the radius of a circle, that will contain A B to any number of sides required.

PROBLEM XXVII.

To make a triangle, whose three sides shall be equal to three given lines D, E, F, if any two are greater than the third.

1. Draw A B equal to the line D.

2. Upon B, with the length of E, describe an arc at C.

3. Upon A, with the length F, describe another arc, intersecting the former at C.

4. Draw A C and C B, and A B C will be the triangle required.

PROBLEM

PROBLEM XXVIII.

To make a trapazium equal, and similar to a given trapazium A B C D.

1. Divide the given trapazium A B C D into two triangles, by a diagonal A C.

2. Make E F equal to A B upon E F, construct the triangle E F, whose three sides will be respectively equal to the triangle A BC.

3. Upon E G, which is equal to A C, construct the triangle EG H, whose two sides E H, and G H, are respectively equal to A D and C D, then E F G H will be the trapazium required.

In the same manner may any irregular polygon be made equal and similar to a given irregular polygon, by dividing the given polygon into triangles, and constructing the triangles in the same manner in the required polygon, as is shown by figures.

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To make a triangle equal to a given trapazium A B C D.

1. Draw the diagonal B D, make C E parallel to it, meeting the side A B, produced in E.

2. Join D E, and A D E will be the triangle.

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To make a triangle equal to any given right-lined figure A B C D E.

1. Produce the side A B both ways at pleasure.

2. Draw the diagonals A D and B D, and make E F and GH parallel to them.

3. Join D F, DG, then D F G will be the triangle required.

Much after the same manner may any other right-line figure be reduced to a triangle.

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