Ex. 365. Prove the above theorem if both lines are tangents.

Ex. 366. Prove the above theorem if one line is a tangent and the other a chord.

Ex. 367. State and prove the converse of the same proposition.

Ex. 368. In the diagram for Prop. XVII, find ABC if arc AB = 80° and arc CD = 120°.

Ex. 369. A line bisecting an angle formed by a tangent and a chord bisects the intercepted arc.

Ex. 370. Prove that Prop. XVII is a limiting case of (225).

229. The opposite angles of an inscribed quadrilateral are supplementary.

B

Hyp. ABCD is an inscribed quadrilateral.

To prove

ZA + ZC = 2 rt. 4.

HINT. — Find the arcs by which A and C are measured

Ex. 371. In the diagram for Prop. XVIII, prove that an exterior angle at C is equal to ▲ A.

Ex. 372. If the opposite angles of a quadrilateral are supplementary, its vertices are concyclic, i.e. a circumference can be described through them.

Ex. 373. Find the sum of three alternate angles of an inscribed hexagon. Ex. 374. The corresponding segments of two equal intersecting chords are equal.

*Ex. 375. If, through the points of intersection of two circumferences, parallels be drawn terminated by the circumferences, these parallels are equal.

Ex. 376. If the bisector of an inscribed angle be produced until it meets the circumference, and through this point of intersection a chord be drawn parallel to one side of the angle, it is equal to the other side.

Ex. 377. If in the greater of two concentric circles, chords be drawn touching the smaller circle, the chords are equal.

Ex. 378. If two equal chords intersect, the lines joining their ends form an isosceles trapezoid.

* Ex. 379. If from the extremities of a diameter perpendiculars be drawn upon any chord (produced, if necessary), the feet of the perpendicular are equidistant from the center.

* Ex. 380. If two unequal chords be produced to meet, the secants thus formed are unequal.

Ex. 381. Which is the shortest line that can be drawn from a point within to a given circumference ? Which is the longest?

* Ex. 382. Each angle formed by joining the feet of the three altitudes of a triangle is bisected by the corresponding altitude.

* Ex. 383. If from any point in the circumference of a circle perpendiculars be dropped upon the sides of an inscribed triangle (produced, if necessary), the feet of the perpendiculars are in a straight line.

* Ex. 384. The tangents drawn at the vertices of an inscribed rectangle enclose a rhombus.

CONSTRUCTIONS

the

230. NOTE. - In the following examples, we shall denote the given parts of a triangle always in the same manner, the sides by a, b, c, opposite angles by A, B, and C, the altitudes by ha, hu, and he, the medians by ma, mb, and me, and the bisectors by ta, to, and to.

PROPOSITION XIX. PROBLEM

231. To construct a triangle, when three sides are given.

ar b

Given. Lines a, b, c.

Required. A triangle having sides equal to a, b, c.

From E as a center, with a radius equal to b, draw an arc. From D as a center, with a radius equal to c, draw an arc. The arcs intersect at F. Join FE and FD.

DISCUSSION.

A DEF is the required triangle.

Q.E.F.

The construction is impossible if one side is greater than the sum of the other two.

Ex. 387. Construct an equilateral triangle, having one side given.
Ex. 388. Construct an equilateral triangle, having given the perimeter.

Ex. 389. To find the third angle of a triangle, when two angles are given.

PROPOSITION XX. PROBLEM

232. To construct a triangle, when one side and two adjacent angles are given.

The solution is left to the student.

Ex. 390. Upon a given base, to construct an isosceles right triangle.

Ex. 391. Construct an isosceles triangle, having given the base and a base angle.

Ex. 392. To construct a quadrilateral, having given the four sides and one angle.

Ex. 393. Is it possible to solve the preceding exercise by constructing first a side not adjacent to the given angle ?

233. REMARK.

The possibility of a solution of a problem depends often upon the proper choice of the part which is drawn first.

PROPOSITION XXI. PROBLEM

234. To construct a triangle, when two sides and an included angle are given.

The solution is left to the student.

Ex. 394. Construct an isosceles triangle, having given an arm and the vertical angle.

Ex. 395. Construct a right triangle, having given the two arms.

PROPOSITION XXII. PROBLEM

235. To construct a triangle, when one side, one adjacent, and one opposite angle are given.

Given. Line a, angles A and B.

Required. A ▲ having ▲ B adjacent to, and ≤ A opposite to,

a side = a.

Construction.

Draw DE equal to a.

At D draw EDF equal to Z B.

At any point, H, in DF, construct

DHG = L A.

Through E, draw a line parallel to HG, meeting DH in I.

DISCUSSION. The construction is impossible if the sum of the given angles is greater than or equal to a straight angle.

Ex. 396. Construct by means of Prop. XXII a right triangle, having given the hypotenuse and an acute angle.

Ex. 397. Find a construction of the same problem which does not depend upon Prop. XXII.

Ex. 398. Construct a right triangle, having given an arm and the opposite angle.

Ex. 399. Construct an equilateral triangle, having given

(a) the altitude.

(b) the radius of the inscribed circle.

(c) the radius of the circumscribed circle.