532. If from a point without a circle two secants be drawn terminating in the concave arc, the product of one secant and its external segment is equal to the product of the other secant and its external segment.

Let AB and BC be two secants drawn from B to the circle whose center is 0.

Prove ▲ AEB and CDB mutually equiangular and similar.

CONVERSE. If on two intersecting lines AB and CB, four points, A, D, C, and E, be taken, so that AB × DB = BC × EB, then can a circumference

esis of the converse, that C can lie neither without nor within

the circumference.]

533. EXERCISE. One of two secants meeting without a circle is 18 in., and its external segment is 4 in. long. The other secant is divided into two equal parts by the circumference. Find the length of the second

secant.

534. EXERCISE. Two secants intersect without the circle. The external segment of the first is 5 ft., and the internal segment 19 ft. long. The internal segment of the second is 7 ft. long. Find the length of each secant.

535. EXERCISE. If A and B are two points such that the polar of A passes through B, then the polar of B passes through A.

536. EXERCISE. The locus of the intersection of tangents to a circle, at the extremities of any chord that passes through a given point, is the polar of the point.

Let CD be any chord passing through A, and B be the point of intersection of the tangents at C and D.

To Prove that B is a point of the polar of A. [CD is the polar of B. (§ 524.)

The polar of B therefore passes through A. By § 535, the polar of A passes through B.]

CE

537. EXERCISE. If from any point on a given line two tangents are drawn to a circle, their chord of contact passes through the pole of the line. [Apply § 535.]

538. EXERCISE. If from different points on a given straight line pairs of tangents are drawn to a circle, their chords of contact all pass through a common point.

PROPOSITION XXIV. THEOREM

539. If from a point without a circle a secant and a tangent are drawn, the secant terminating in the concave arc, the square of the tangent is equal to the product of the secant and its external segment.

Let AB be a tangent and BC a secant drawn from B to the circle whose center is 0.

PROPOSITION XXV. THEOREM

542. Two polygons are similar if they are composed of the same number of triangles, similar each to each, and similarly placed.

Let the AABC, ADC, DEC, and EFC be similar respectively to the AGHI, GJI, JLI, and LMI, and be similarly placed.

To Prove polygons ABCFED and GHIMLJ similar.

Proof. Show that the angles of ABCFED are equal respectively to the corresponding angles of GHIMLJ.

GH GJ JL LM MI IH

The polygons are mutually equiangular and have their corresponding sides proportional. They are therefore similar by definition.

Q.E.D.

543. COROLLARY.

On a given line to construct a polygon

similar to a given polygon.

544. DEFINITION. In similar polygons the corresponding sides are called homologous sides, and the equal angles are called homologous angles.

PROPOSITION XXVI. THEOREM

545. Two similar polygons can be divided into the same number of similar triangles, similarly placed.

Let ABCDEF and GHIJLM be two similar polygons.

To Prove that they can be divided into the same number of similar triangles, similarly placed.

Proof.

From the vertex F draw all the possible diagonals. From M, homologous with F, draw all the possible diagonals. Prove▲ FAB and MGH similar (§ 495).

A FBC and MIII are similar. (?)

Show that FCD and MIJ are similar, and also ▲ FDE and

MJL.

Q.E.D.