57. To find a point in a straight line such that the angle subtended thereat by the straight line joining two given points on the same side of it is a maximum.

B

A

E

Let CD be the given straight line, and A, B the given points.

Describe a circle through A, B, touching CD at C (Prob. 56).

C is the point required.

In other words, if E be any other point in CD, the angle ACB is greater than the angle AEB.

Proof. The angle ACB the angle AFB, for they are in the same segment.

And the angle AFB is greater than the angle AEB (Euc. I. 16).

.. &c.

F

58. To find a point in a straight line such that the sum of its distances from two fixed points on the same side of the line is a minimum.

B

Let A, B, be the given points, and CP the given straight line.

Draw AC perpendicular to CP, and make CD = CA.

Join BD meeting CP in P.

P is the point required.

Proof. For take Q any other point in CP, and join AQ BQ.

Then AP BP will be less than AQ + BQ.

For by applying Euc I. 4, we find DP AP and DQ AQ.

Hence AP BP DB, which is less than DQ+ BQ (Euc. I. 20); and therefore AP + BP is less than AQ + BQ.

59. To inscribe a circle in a given sector of a circle.

B

E

Let ABC be the given sector.

Bisect the angle at A by AD, meeting the arc in D.

Draw DE at right angles to AD.

Bisect the angle AED by EF, meeting AD in F.

With center F and radius FD describe a circle.

This is the circle required.

Note. This problem is evidently identical with that by which we inscribe a circle in a given triangle AEF.

60. To inscribe any number (say five) of equal circles in a given circle.

B

Divide the circle into five equal sectors, and bisect the angles at the center, O.

Inscribe a circle in one sector, OAE (by Prob. 59).

Let the center of this be H.

Mark off from O on the bisectors of the other sectors distances equal to OH.

These will be the centers of the remaining circles, and the radius of each will be equal to HG.

61. To describe any number of equal circles (say eight) about a given circle, each circle to touch two others besides the given circle.

Divide the given circle into eight equal sectors, of which let AOB be one.

Bisect the sector AOB by OD, meeting the arc in D.

Draw DC at right angles to OD.

Bisect the angle DCF by CE, meeting OD produced in E.

With center E and radius ED describe a circle.

This is one of the eight circles required.

Bisect the other sectors, and mark off on these bisectors, measuring from O, distances equal to OE.