PROBLEM I.

In a right angled triangle BAC, having given the base BA, and the sum of the hypothenuse and perpendicular, it is required to find the hypothenuse and perpendicular.

Put BA=c=3, BC=x, AC=y and the sum of the hypothenuse and perpendicular equal to

Then,

x+y=s=9.

S=9

and x2=y2+c2 (Bk. IV. Prop. XI.) From 1st equ: x=s-y

and

x2=s2-2sy + y2

B

In a right angled triangle, having given the hypothenuse, and the sum of the base and perpendicular, to find these two sides

By completing the square y-sy + } s2 = }} a2-1s2

or

Hence

y=¿s±√4a2—¡s2=4 or 3

x=!s=√√§a°—4s2=3 or 4

PROBLEM III.

In a rectangle, having given the diagonal and perimeter, to find

the sides.

Let ABCD be the proposed rectangle. Put AC=d=10, the perimeter=2a=28, or AB+BC=a=14: also put AB: =x and BC=y.

From which equations we obtain,

y=}a± √ } d2 a2-8 or 6,

and

x=}a= √{d2—}a2=6 or 8.

PROBLEM IV.

Having given the base and perpendicular of a triangle, to find

the side of an inscribed

square.

which, therefore, depends only on the base and altitude of the triangle.

PROBLEM V.

In an equilateral triangle, having given the lengths of the three perpendiculars drawn from a point within, on the three sides: to determine the sides of the triangle.

S*

Let ABC be the equilateral triangle; DG, DE and DF the given perpendiculars let fall from D on the sides. Draw DA, DB, DC to the vertices of the angles, and let fall the perpendicular CH on the base. Let DG=a, DE=b, and DF-c: put one of the equal sides AB

F

E

H G B

=2x; hence AH=x, and CH=√AC2_AH2=√√/4x2—x2

Now since the area of a triangle is equal to half its base into the altitude, (Bk. IV. Prop. VI.)

AB × CH=x × ≈ √3=x2 √3=triangle ACB

ABXDG=xx a =αx

=triangle ADB BCXDE=xx b =bx triangle BCD

ACX DF=xxc

=Cx =triangle ACD

But the three last triangles make up, and are consequently equal to, the first; hence,

REMARK. Since the perpendicular CH is equal to x√√3, it is consequently equal to a+b+c: that is, the perpendicular let fall from either angle of an equilateral triangle on the opposite side, is equal to the sum of the three perpendiculars let fall from any point within the triangle on the sides respectively.

PROBLEM VI.

In a right angled triangle, having given the base and the difference between the hypothenuse and perpendicular, to find the sides.

PROBLEM VII.

In a right angled triangle, having given the hypothenuse and the difference between the base and perpendicular, to determine the triangle.

PROBLEM VIII.

Having given the area of a rectangle inscribed in a given triangle; to determine the sides of the rectangle.

PROBLEM IX.

In a triangle, having given the ratio of the two sides, together with both the segments of the base made by a perpendicular from the vertical angle; to determine the triangle.

PROBLEM X.

In a triangle, having given the base, the sum of the other two sides, and the length of a line drawn from the vertical angle to the middle of the base; to find the sides of the triangle.

PROBLEM XI.

In a triangle, having given the two sides about the vertical angle, together with the line bisecting that angle and terminating in the base; to find the base.

PROBLEM XII.

To determine a right angled triangle, having given the lengths of two lines drawn from the acute angles to the middle of the opposite sides.

PROBLEM XIII.

To determine a right-angled triangle, having given the pcrimeter and the radius of the inscribed circle.

PROBLEM XIV.

To determine a triangle, having given the base, the perpendicular and the ratio of the two sides.

PROBLEM XV.

To determine a right angled triangle, having given the hypothenuse, and the side of the inscribed square.

PROBLEM XVI.

To determine the radii of three equal circles, described within and tangent to, a given circle, and also tangent to each other.

PROBLEM XVII

In a right angled triangle, having given the perimeter and the perpendicular let fall from the right angle on the hypothenuse, to determine the triangle.

PROBLEM XVIII.

To determine a right angled triangle, having given the hypothenuse and the difference of two lines drawn from the two acute angles to the centre of the inscribed circle.

PROBLEM XIX.

To determine a triangle, having given the base, the perpendicular, and the difference of the two other sides.

PROBLEM XX.

To determine a triangle, having given the base, the perpendicular and the rectangle of the two sides.

PROBLEM XXI.

To determine a triangle, having given the lengths of three lines drawn from the three angles to the middle of the opposite sides.

PROBLEM XXII.

In a triangle, having given the three sides, to find the radius of the inscribed circle.

PROBLEM XXIII.

To determine a right angled triangle, having given the side of the inscribed square, and the radius of the inscribed circle.

PROBLEM XXIV.

To determine a right angled triangle, having given the hypothenuse and radius of the inscribed circle.

PROBLEM XXV.

To determine a triangle, having given the base, the line bisecting the vertical angle, and the diameter of the circumscribing circle.