THEOREM XII.

Two parallel straight lines intercept equal arcs on a circumference.

THEOREM XIII.

In the same circle or in equal circles, two angles at the centre are in the same ratio as their intercepted arcs.

THEOREM XIV.

An angle inscribed in a circumference is measured by one half its intercepted arc.

Corollary. right angle.

An angle inscribed in a semi-circumference is a

THEOREM XV.

If a quadrilateral is inscribed in a circle, the sum of two opposite angles is two right angles; and conversely, if the sum of two opposite angles of a quadrilateral is two right angles. the quadrilateral can be inscribed in a circle.

THEOREM XVI.

An angle formed by a tangent and a chord is measured by one half the intercepted arc.

THEOREM XVII.

An angle formed by two chords intersecting each other within a circumference is measured by one half the sum of the arcs intercepted between its sides and between the sides of its vertical angle.

THEOREM XVIII.

An angle formed by two secants intersecting each other without a circumference, by two tangents, or by a tangent and a secant, is measured by one half the difference of the intercepted

arcs.

BOOK III.

SIMILAR POLYGONS.

THEOREM I.

A straight line parallel to the base of a triangle divides the other two sides proportionally.

Conversely, if a straight line divides two sides of a triangle proportionally, it is parallel to the third side.

THEOREM II.

If two triangles have their angles respectively equal, the triangles are similar.

Corollary. If two triangles have two angles of one respectively equal to two angles of the other, the triangles are similar.

THEOREM III.

If two triangles have an angle of one equal to an angle of the other and the sides including these angles proportional, the triangles are similar.

THEOREM IV.

If two triangles have their sides respectively proportional, the triangles are similar.

THEOREM V.

The bisector of an angle of a triangle divides the opposite side into segments proportional to the sides of the angle.

THEOREM VI.

If two polygons are composed of the same number of triangles, similar each to each and similarly placed, the polygons are similar.

Conversely, if two polygons are similar, they can be decomposed into the same number of triangles, similar each to each and similarly placed.

THEOREM VII.

The perimeters of two similar polygons are in the same ratio as any two corresponding sides.

THEOREM VIII.

If in a right triangle a perpendicular is drawn from the vertex of the right angle to the hypotenuse:

I. The two triangles thus formed are similar to each other and to the whole triangle.

II. The perpendicular is a mean proportional between the segments of the hypotenuse.

III. Each leg of the right triangle is a mean proportional between the hypotenuse and the segment adjacent to that leg.

THEOREM IX.

The product of the segments of a chord that passes through a fixed point within a circle is the same for all directions of the chord.

THEOREM X.

If from a fixed point without a circle a secant is drawn, the product of the whole secant and its external segment is the same for all directions of the secant.

Corollary. If a tangent and a secant intersect, the tangent is a mean proportional between the whole secant and its external segment.

BOOK IV.

AREAS OF POLYGONS.

THEOREM I.

Parallelograms having equal bases and equal altitudes are equivalent.

THEOREM II.

The areas of two rectangles having equal altitudes are to each other as their bases.

THEOREM III.

The areas of two rectangles are to each other as the products of their bases and their altitudes.

Corollary. The area of a rectangle is equal to the product of its base and its altitude.

THEOREM IV.

The area of a parallelogram is equal to the product of its base and its altitude.

THEOREM V.

The area of a triangle is equal to half the product of its base and its altitude.

Corollary. Two triangles having equal bases and equal altitudes are equivalent.

THEOREM VI.

The area of a trapezoid is equal to the product of its altitude and half the sum of its parallel sides.

THEOREM VII.

The areas of two similar triangles are to each other as the squares of any two corresponding sides.

Corollary. The areas of two similar polygons are to each other as the squares of any two corresponding sides; and also as the squares of their perimeters.

THEOREM VIII.

The square described on the hypotenuse of a right triangle is equivalent to the sum of the squares described on the other two sides.