PROPOSITION XXXII.-PROBLEM.

80. To inscribe a circle in a given triangle.

F

Let ABC be the given triangle. Bisect any two of its angles, as B and C, by straight lines meeting in O. From the point O let fall perpendiculars OD, OE, OF, upon the three sides of the triangle; these perpendiculars will be equal to each other (I., Proposition XIX.). Hence the circumference of B a circle, described with the centre O, and

D

E

a radius = OD, will pass through the three points D, E, F, will be tangent to the three sides of the triangle at these points (Proposition IX.), and will therefore be inscribed in the triangle.

EXERCISE.

Problem.-Upon a given straight line, to describe a segment which shall contain a given angle.

Suggestion. Through one end of the given line AB draw a line BC, making with it the given angle. The two lines will be one a chord and the other a tangent. Hence the centre of the circle can be found.

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A

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EXERCISES ON BOOK II.

THEOREMS.

1 IF two circumferences are tangent internally, and the radius of the larger is the diameter of the smaller, then any chord of the larger drawn from the point of contact is bisected by the circumference of the smaller (v. Proposition XIV., Corollary, and Proposition VI.).

2. If two equal chords intersect within a circle, the segments of one are respectively equal to the segments of the other. What is the corresponding theorem for the case where the chords meet when produced?

3. A circumference described on the hypotenuse of a right triangle as a diameter passes through the vertex of the right angle. (v. Proposition XIV., Corollary.)

4. The circles described on two sides of a triangle as diameters intersect on the third side.

Suggestion. Drop a perpendicular from the opposite vertex upon the third side.

5. The perpendiculars from the angles upon the opposite sides of a triangle are the bisectors of the angles of the triangle formed by joining the feet of the perpendiculars.

Suggestion. On the three sides of the given triangle as diameters describe circumferences. (v. Exercise 3, Proposition XIV., and I., Proposition XXVI.).

6. If a circle is circumscribed about an equilateral triangle, the perpendicular from its centre upon a side of the triangle is equal to one-half of the radius.

7. The portions of any straight line which are intercepted between the circumferences of two concentric circles are equal.

9. If a triangle ABC is formed by the intersection of three

tangents to a circumference, two of which, AM and AN, are fixed, while the third, BC, touches the circumference at a variable point P, prove that the perimeter of the triangle ABC is constant, and equal to AM + AN, or 2AN (Proposition X.).

Also, prove that the angle BOC is constant.

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10. If through one of the points of intersection of two circumferences a diameter of each circle is drawn, the straight line which joins the extremities of these diameters passes through the other point of intersection, and is parallel to the line joining the centres.

Suggestion. Draw the common chord and the line joining the centres. (v. Proposition VI., Corollary II., and Exercise 29, Book I.)

12. A circle can be entirely surrounded by six circles having the same radius with it.

13. The bisectors of the vertical angles of all triangles having the same base and equal vertical angles have a point in common. Suggestion. The triangles may all be inscribed in the same

circle.

14. If the hypotenuse of a right triangle is double one of the sides, the acute angles of the triangle are 30° and 60° respectively. 15. If, from a point whose distance from the centre of a given circle is equal to a diameter, tangents are drawn to the circle, they will make with each other an angle of 60o.

LOCI.

16. Find the locus of the centre of a circumference which passes through two given points. (v. I., Proposition XVIII.)

17. Find the locus of the centre of a circumference which is tangent to two given straight lines. (v. I., Proposition XIX.)

18. Find the locus of the centre of a circumference which is tangent to a given straight line at a given point of that line, or to a given circumference at a given point of that circumference.

19. Find the locus of the centre of a circumference passing through a given point and having a given radius.

20. Find the locus of the centre of a circumference tangent to a given straight line and having a given radius.

21. Find the locus of the centre of a circumference of given radius, tangent externally or internally to a given circumference.

22. A straight line MN, of given length, is placed with its extremities on two given perpendicular lines AB, CD; find A the locus of its middle point P (Exercise 31, Book I.).

23. A straight line of given length is inscribed in a given circle; find the locus of its middle point. (v. Proposition VII.)

24. A straight line is drawn through a given point A, intersecting a given circumference in B and C; find the locus of the middle point, P, of the intercepted chord BC.

Note the special case in which the point A is on the given circumference.

25. From any point A in a given circumference, a straight line AP of fixed length is drawn parallel to a given line MN; find the locus of the extremity P. (v. I., Proposition XXX.)

26. From one extremity A of a fixed diameter AB, any chord AC is drawn, and at C a tangent CD. From B, a perpendicular BD to the tangent is drawn, meeting AC in P. Find the locus of P. Suggestion. (Draw radius OC. v. I., Exercise 28.)

27. The base BC of a triangle is given, and the medial line BE, from B, is of a given length. Find the locus of the vertex A.

Suggestion. Draw AO parallel to EB. Since BO = BC, O is a fixed point; and since AO = 2BE, OA is a constant distance.