are equal and the circle described from the centre G, and distance of any one of them, will pass through the extremities of the other two, and touch the arc and sides in the points D, F, H, because the angles at those points are right angles.

(32.) To describe a circle, the circumference of which shall pass through a given point, and touch a given straight line in a given point.

Let AB be the given straight line, C the given point, in which the circle is to touch it, D the point through which it must pass. Draw CO perpendicular to

A

AB. Join CD; and at the point D make the angle CDO=DCO; the intersection of the lines CO and DO is the centre of the circle required.

Since the angle DCO=CDO, CO=DO, and .`. a circle described from the centre 0, at the distance OD, will pass through C, and touch the line AB in C, bebecause OC is perpendicular to AB.

(33.) To describe a circle which shall pass through a given point, have a given radius, and touch a given straight line.

Let AB be the given straight line, and C the given point through which the circle must pass.

In AB take any point B; and from

it draw BD at right angles to AB, and equal to the given radius; through D draw DE parallel to AB; and with the centre C, and radius equal to the given radius, describe a circle cutting DE in O. O is the centre of the circle required.

From O draw OF perpendicular to AB, it is equal to DB, i. e. to the given radius; and the circle described from the centre O, and radius OF, will touch (Eucl. iii. 16. Cor.) the line AB in F, and pass through C.

(34.) To describe a circle which shall pass through two given points, and touch a given straight line.

Let A, B be the given points, and CD the given straight line. Join AB. And 1. let CD be parallel to AB.

Bisect AB in E, and draw EF perpen

F

E

B

dicular to AB, and .. to CD. Join FA, and make the angle FAO AFO; then will O be the centre of the circle required.

=

Since the angle FAO=AFO, A0=OF. But AE= EB, and EO is common to the triangles AEO, BEO, and the angles at E right angles, .. AO=OB. Whence AO, OB, OF are all equal; and the circle described from the centre O, at the distance of any one of them, will pass through the extremities of the other two, and touch the line CD, since OF is perpendicular to CD.

2. But if AB is not parallel to CD, let them be produced to meet in E; and take EF a mean proportional between EA and EB. Join FA, FB; and describe a circle about the triangle AFB; it will be the circle required.

F

B

Since EF is a mean proportional between EA and EB, EF touches the circle (Eucl. iii. 37.), which passes through A and B.

(35.) To describe a circle, the circumference of which shall pass through a given point, and touch a circle in a given point; the two points not being in a tangent to the given circle.

Let A be the given point in the circumference of the circle whose centre F is O; B the given point without. Join BA, and produce it to D. Join OD; and through A draw OAE; and draw

E

B

BE parallel to OD, cutting OAE in E. E is the centre of the circle required.

Since (Eucl. i. 29.) the angle ODA is equal to ABE, and OAD to BAE, .. the triangles ODA, ABE are similar, and OD being equal to OA, AE will be equal to EB; a circle .. described with the centre E, and radius EA, will pass through B, and touch the circle ADF in the point A, since the line joining the centres passes through A.

(36.) To describe a circle the centre of which may be in the perpendicular of a given right-angled triangle, and the circumference pass through the right angle and touch the hypothenuse.

Let EAD be the given right-angled triangle, having the angle at A a right angle. Make EC EA. Join

=

CA; and draw CO at right angles to ED. The circle described with O as centre, and radius OA, will be the circle required.

=

Since EA EC, the angle ECA= EAC, and ECO, EAO are equal, being right angles; .. OCA OAC and OA=

=

OC. The circle .. described from the centre O, and radius OA, will pass through the extremity of OC, and touch ED in C, because CO is at right angles to ED.

(37.) To describe a circle which shall pass through the extremities of a given line, so that if from any point in its circumference a line be drawn making a given angle with the given line; the rectangle contained by the segment it cuts off and the given line, may be equal to the square of the line drawn from the same point to the farther extremity of the given line.

Let AB be the given line. On it describe a segment of a circle containing an angle equal to the given angle. Complete the circle; it will be the one required.

A

BD

From any point C draw CD, making with AB the angle ADC equal to the given angle; join CA, CB. Since the angle CDA= ACB, and the angle at A is common, the triangles ACD, ABC are equiangular, and therefore

AB AC :: AC : AD,

whence the rectangle contained by AB, AD, is equal to the square of AC.

(38.) To determine a point in the perpendicular let fall from the vertical angle of a triangle on the base; about which as a centre a circle may be described touching the longer side, and passing through the opposite angular point.

Let ABC be a triangle, and from B the vertex let BD be drawn perpendicular to AC. In DB take any point E, and from it draw EF perpendicular to AB; and from E to BC, draw EG=EF;

B

F

E

H

D

from C draw CH parallel to GE, and from H draw Hl perpendicular to AB; H is the point required.

Since EF is parallel to HI,

FE: HI :: BE : BH,

and since GE is parallel to HC,

GE

HC: BE : BH,

.. FE HI :: GE HC;

but, by construction, FE = EG, .. HI=HC; and a circle described from the centre H at the distance HI, will pass through C, and touch AB in I, since the angle HIB is a right angle.

(39.) To describe a circle which shall have a given radius, and its centre in a given straight line, and shall also touch another given straight line inclined at a given angle to the former.

Let AB be the given line, in which the centre is to be; BC the line which the circle is to touch.

In BC take any point C, and draw CD at right angles to it; and make

C

D

B