180. In the figure on page 73,

NOTE.

1st. If G F and K E are drawn, the angle

FGA+AFG+CKE+KE C= a right angle.

Letters once written on the figure, or lines drawn, are assumed

as still remaining in the subsequent propositions.

2d. The lines joining A H and CI are parallel.

3d. G, B, and K are in the same straight line.

4th. If perpendiculars are drawn from G and K to A C produced, BD is a mean proportional between them; the parts of AC produced will be equal, and the perpendiculars together will be equal to A C.

5th. Draw BE. BF2-BE2

=

A B2 B C2.

6th. If perpendiculars are drawn from F and E to GA and
K C, cutting them in L and M respectively, the triangles
FAL and E M C will each be equal to A B C.

7th. If FA and E C are produced to meet G H and K I in
N and O respectively, the triangle A G N = COK.
8th. If G H and K I are produced to meet in P, N P = A B
and P 0=B C.

9th. If H I and NO are drawn, the triangle HPI=NPO.
10th. The sum of the perpendiculars from H and I to A C
AC+2 BD.

=

11th. If the perpendiculars from H and I meet A C in Q and R respectively, then HIRQ=2 (B D2 + A B C). 12th. Draw A K. BD passes through the intersection of G C and A K.

13th. If G C cuts A B in T, and H T is drawn, the triangle

16th. If TD, D S are drawn, the angle TDS is a right angle. 17th. A T: TB = TB: SC.

22d. If G A and K L are produced to meet in U, then G P KU

is a square.

23d. L, S, and I are in the same straight line.

24th. A K is perpendicular to BE; G C to F B.

25th. H T is parallel to B E.

26th. If L I is drawn, then BILF is a parallelogram, and

2 ABC.

=

27th. If B F and BE cut AC in W and Y respectively,

then AW: WY WY: Y C.

=

28th. AT: SC AW: YC =

=

AD: DC, and A T: TB,

BS: SC, TD: DS as AB: BC.

29th. If A K cuts B F in a, and G C cuts BE in b,

31st. If Z is the point in which AK and G C intersect BD

(12th), W D: DZDZ: DY.

32d. If W Z, Z Y are drawn, WZ Y is a right angle, and W Z is parallel to A B, and Z Y to B C.

35th. A Z XZ C = BW× BY.

36th. From W and Y erect perpendiculars to A C, meeting A B, B C in c, d respectively, and join cd; then Wcd Y will be a square.

37th. From T draw a line parallel to B C, meeting A C in e, and join e S; then TB Se is a square.

1

40th. The sum of the squares on the sides of the polygon FGHIKE is equal to eight times the square on A C. 41st. Let A K cut B E in f, and G C cut B A in g ;

BOOK III.

93. If two circumferences touch each other, the chords of each forming a straight line through the point of contact have a constant ratio.

94. If of two circles the diameter of one is the radius of the other, any chord of the first drawn from the point of contact of the two circumferences is bisected by the circumference of the second.

95. If two circumferences cut each other, the extremities of their diameters drawn from one of the points of intersection are in the same straight line with the other point of intersection.

96. If A B and A C are tangents to a circumference from A, and DE is tangent to the circumference at any point between B and C, D being a point in A B, and E in A C, the perimeter of the triangle ADE is constant.

97. Prove I. 144 by principles in Book III.

98. The difference between the sum of the two sides of a right triangle and the hypothenuse is equal to the diameter of the inscribed circle.

99. If lines are drawn from any point of the circumference of a circle to the vertices of an inscribed equilateral triangle, the middle line is equal to the sum of the other two.

100. The lines bisecting the angles formed by producing the opposite sides of a quadrilateral inscribed in a circle intersect at right angles.

101. The rectangle contained by two sides of a triangle is equal to the rectangle contained by the segments of the third side made by a line bisecting the opposite angle, plus the square of the bisecting line.

(Circumscribe a circle about the triangle, produce the bisecting line beyond the side to the circumference, and join the point where it meets the circumference with one extremity of this side.)

102. The rectangle contained by two sides of a triangle is equivalent to the rectangle contained by the diameter of the circumscribed circle and the perpendicular upon the third side from the vertex of the opposite angle.

103. The area of a triangle is equal to the product of its three sides divided by twice the diameter of the circumscribed circle.

104. The rectangle contained by the diagonals of a quadrilateral inscribed in a circle is equal to the sum of the two rectangles contained by the opposite sides.

105. If a perpendicular is drawn from the vertex of a triangle A B C, to the base A C, the base is to the sum of the other two sides as the difference of these sides is to the difference of the segments of the base.

(With B as a centre, and the shorter of the two sides A B, B.C as a radius, describe a circle; produce CB to meet the circumference; if the perpendicular falls without the triangle, in like manner produce C A.)

ure.

106. The diagonal and side of a square have no common meas

107. Prove I. 143, first circumscribing a circle about the triangle and producing one of the perpendiculars to meet the circumference, by reference to III. 24.

108. Prove II. 66, first drawing a circle with the vertex of one of the acute angles as a centre, and the adjacent side as a radius, and from the vertex of the right angle drawing chords to the points where the circumference cuts the hypothenuse and the hypothenuse produced.

109. If a circle is circumscribed about a right triangle, and on each of the sides including the right angle as diameters, semicircles are described without the triangle, the sum of the areas of the crescents thus formed is equal to the area of the right triangle.

110. In the figure used in II. 180, T, B, S, D, and e are all in one circumference.

BOOK IV.

25. The difference of two lines drawn to a point in a straight line from points on opposite sides of this line is a maximum when these lines make equal angles with the given line.

26. Of parallelograms with sides mutually equal the maximum is rectangular.

27. The sum of the squares of two lines is never less than twice their rectangle.

28. The sum of the squares of the perpendiculars from a point within a rectangle to the sides is a minimum when this point is the centre of the rectangle.

29. The perimeter of an isosceles triangle is greater than that of an equal rectangle of the same altitude.

30. The sum of the triangles cut off by lines drawn from a point in the base of an isosceles triangle parallel to the sides, is the minimum when the point is at the centre of the base.

31. Of all triangles that have the same vertical angle and whose bases pass through a given point, the minimum is the one whose base is bisected at the given point.

32. Of all the squares that can be inscribed in a given square the minimum has its vertices at the middle points of the sides.

33. Of all the triangles whose vertices are on the sides of a given triangle the perimeter of the one that has its vertices at the feet of the perpendiculars from the vertices of the given triangle to the opposite sides, is the minimum.