Obs. In examples on the Intersection of Loci the student should make a point of investigating the relations which must exist among the data, in order that the problem may be possible; and he must observe that if under certain relations two solutions are possible, and under other relations no solution exists, there will always be some intermediate relation under which the two solutions combine in a singie solution.

EXAMPLES ON LOCI.

1. Find the locus of a point which moves so that its distance (measured radially) from the circumference of a given circle is constant.

2. A point P moves along a straight line RQ; find the position in which it is equidistant from two given points A and B.

3. A and B are two fixed points within a circle: find points on the circumference equidistant from A and B. How many such points are there?

4. A point P moves along a straight line RQ; find the position in which it is equidistant from two given straight lines AB and CD.

5. A and B are two fixed points 6 cm. apart. Find by the method of loci two points which are 4 cm. distant from A, and 5 cm. from B.

6. AB and CD are two given straight lines. Find points 3 cm. distant from AB, and 4 cm. from CD. How many solutions are there?

7. A straight rod of given length slides between two straight rulers placed at right angles to one another.

Plot the locus of its middle point; and shew that this locus is the fourth part of the circumference of a circle. [See Problem 10.]

8. On a given base as hypotenuse right-angled triangles are described. Find the locus of their vertices.

9.

A is a fixed point, and the point X moves on a fixed straight line BC.

Plot the locus of P, the middle point of AX; and prove the locus to be a straight line parallel to BC.

10. A is a fixed point, and the point X moves on the circumference of a given circle.

Plot the locus of P, the middle point of AX; and prove that this locus is a circle. [See Ex. 3, p. 64.]

11. AB is a given straight line, and AX is the perpendicular drawn from A to any straight line passing through B. If BX revolve about B, find the locus of the middle point of AX.

12.

a

12. Two straight lines OX, OY cut at right angles, and from P, point within the angle XOY, perpendiculars PM, PN are drawn to OX, OY respectively. Plot the locus of P when

(i) PM + PN is constant (=6 cm., say);

(ii) PM - PN is constant (=3 cm., say).

And in each case give a theoretical proof of the result you arrive at experimentally.

13. Two straight lines OX, OY intersect at right angles at O; and from a movable point P perpendiculars PM, PN are drawn to OX, OY. Plot (without proof) the locus of P, when

(i) PM 2 PN;

(ii) PM 3 PN.

14. Find a point which is at a given distance from a given point, and is equidistant from two given parallel straight lines.

When does this problem admit of two solutions, when of one only, and when is it impossible?

15. S is a fixed point 2 inches distant from a given straight line MX. Find two points which are 2 inches distant from S, and also 22 inches distant from MX.

16. Find a series of points equidistant from a given point S and a given straight line MX. Draw a curve freehand passing through all

the points so found.

17. On a given base construct a triangle of given altitude, having its vertex on a given straight line.

18. Find a point equidistant from the three sides of a triangle.

19. Two straight lines OX, OY cut at right angles; and Q and R are points in OX and OY respectively. Plot the locus of the middle point of QR, when

(i) OQ+OR= constant.

(ii) OQ-OR=constant.

20. S and S′ are two fixed points. Find a series of points P such that

(i) SP+S'P constant (say 3.5 inches).

(ii) SP-S'P= constant (say 1.5 inch).

In each case draw a curve freehand passing through all the points BO found.

ON THE CONCURRENCE OF STRAIGHT LINES IN A TRIANGLE.

I.

The perpendiculars drawn to the sides of a triangle from their middle points are concurrent.

Let ABC be a A, and X, Y, Z the middle points of its sides.

From Z and Y draw perps. to AB, AC, meeting at O. Join OX.

It is required to prove that OX is perp. to BC.

Proof.

Because YO bisects AC at right angles,

.. it is the locus of points equidistant from A and C ;
.. OA=OC.

Again, because ZO bisects AB at right angles,

.. it is the locus of points equidistant from A and B ;
.. OA OB.

Hence OB OC.

.. O is on the locus of points equidistant from B and C :
that is, OX is perp. to BC.

Hence the perpendiculars from the mid-points of the sides meet at 0.

Q.E.D.

II. The bisectors of the angles of a triangle are concurrent.

Let ABC be a A. Bisect the ABC, BCA by straight lines which meet at O.

Join AO.

It is required to prove that AO bisects the L BAC.

From O draw OP, OQ, OR perp. to the sides of the A.

Proof.

B

Because BO bisects the ABC,

P

.. it is the locus of points equidistant from BA and BC ;

.. OP OR.

Similarly CO is the locus of points equidistant from BC and CA; .. OP=OQ.

Hence OR OQ.

.. O is on the locus of points equidistant from AB and AC;
that is, OA is the bisector of the L BAC.

Hence the bisectors of the angles meet at O.

Q.E.D.

because Y is the middle point of AC, and YO is parallel to CK,

.. O is the middle point of AK.

Again in the ▲ ABK,

since Z and O are the middle points of AB, AK,
.. ZO is parallel to BK,

that is, OC is parallel to BK,
.. the figure BKCO is a par".

But the diagonals of a parm bisect one another;
.. X is the middle point of BC.

That is, AX is a median of the A.

Hence the three medians meet at the point O.

Theor. 22.

Q.E.D.

DEFINITION. The point of intersection of the medians is called the centroid of the triangle.

COROLLARY. The three medians of a triangle cut one another at a point of trisection, the greater segment in each being towards the angular point.

For in the above figure it has been proved that

AO = OK,

also that OX is half of OK;

.. OX is half of OA :

that is, OX is one third of AX.

Similarly OY is one third of BY,

and OZ is one third of CZ.

Q.E.D.

By means of this Corollary it may be shewn that in any triangle the shorter median bisects the greater side.

NOTE. It will be proved hereafter that the perpendiculars drawn from the vertices of a triangle to the opposite sides are concurrent.

MISCELLANEOUS PROBLEMS.

(A theoretical proof is to be given in each case.)

1. A is a given point, and BC a given straight line. From A draw a straight line to make with BC an angle equal to a given angle X. How many such lines can be drawn?

j 2. Draw the bisector of an angle AOB, without using the vertex O in your construction.

3. P is a given point within the angle AOB. Draw through Pa straight line terminated by OA and OB, and bisected at P.

4. OA, OB, OC are three straight lines meeting at O. transversal terminated by OA and OC, and bisected by OB.

Draw a

5. Through a given point A draw a straight line so that the part intercepted between two given parallels may be of given length.

When does this problem admit of two solutions? When of only one? And when is it impossible?

6. In a triangle ABC inscribe a rhombus having one of its angles coinciding with the angle A.

7. Use the properties of an equilateral triangle to trisect a given straight line.

(Construction of Triangles.)

8. Construct a triangle, having given

(i) The middle points of the three sides.

(ii) The lengths of two sides and of the median which bisects the third side.

(iii) The lengths of one side and the medians which bisect the other two sides.

(iv) The lengths of the three medians.